YALE  UNIVERSITY 
MRS.  HEPSA  ELY  SILLIMAN  MEMORIAL  LECTURES 


EXPERIMENTAL  AND   THEORETICAL 

APPLICATIONS    OF 
THERMODYNAMICS   TO   CHEMISTRY 


SILLIMAN  MEMORIAL  LECTURES 

PUBLISHED  BY  CHARLES  SCRIBNER'S  SONS 


ELECTRICITY  AND  MATTER.  By  Professor  J. 
J.  Thomson.  Net,  $1.25. 

THE   INTEGRATIVE   ACTION   OF   THE    NERVOUS 

SYSTEM.     By  Professor  C.  S.  Sherring- 
ton.    Net,  $3.50. 

RADIOACTIVE  TRANSFORMATIONS.  By  Pro- 
fessor E.  Rutherford.  Nett  $3.50. 

EXPERIMENTAL  AND  THEORETICAL  APPLICA- 
TIONS OF  THERMODYNAMICS  TO  CHEM- 
ISTRY. By  Professor  Walther  Nernst. 
Net,  $1.25. 


EXPERIMENTAL  AND 
THEORETICAL  APPLICATIONS   OF 

THERMODYNAMICS 
TO    CHEMISTRY 


BY 


DR.   WALTHER    NERNST 

ii 

FROFESSOR   AND  DIRECTOR   OF  THE   INSTITUTE  OF   PHYSICAL 
CHEMISTRY   IN   THE   UNIVERSITY  OF   BERLIN 


WITH    DIAGRAMS 


NEW   YORK 

CHARLES    SCRIBNER'S   SONS 
1907 


SEVERAL 


COPYRIGHT,  1907 
BY  YALE  UNIVERSITY 

Published,  April,  1907 


TROW   DIRECTORY 

PRINTING  AND   BOOKBINDING  COMPAN 
NEW  YORK 


THE  SILLIMAN  FOUNDATION 

IN  the  year  1883  a  legacy  of  eighty  thousand  dollars 
was  left  to  the  President  and  Fellows  of  Yale  College 
in  the  City  of  New  Haven,  to  be  held  hi  trust,  as  a 
gift  from  her  children,  in  memory  of  their  beloved  and 
honored  mother  Mrs.  Hepsa  Ely  Silliman. 

On  this  foundation  Yale  College  was  requested  and 
directed  to  establish  an  annual  course  of  lectures  de- 
signed to  illustrate  the  presence  and  providence,  the 
wisdom  and  goodness  of  God,  as  manifested  in  the 
natural  and  moral  world.  These  were  to  be  designated 
as  the  Mrs.  Hepsa  Ely  Silliman  Memorial  Lectures.  It 
was  the  belief  of  the  testator  that  any  orderly  presenta- 
tion of  the  facts  of  nature  or  history  contributed  to 
the  end  of  this  foundation  more  effectively  than  any 
attempt  to  emphasize  the  elements  of  doctrine  or  of 
creed ;  and  he  therefore  provided  that  lectures  on  dog- 
matic or  polemical  theology  should  be  excluded  from 
the  scope  of  this  foundation,  and  that  the  subjects  should 
be  selected  rather  from  the  domains  of  natural  science 
and  history,  giving  special  prominence  to  astronomy, 
chemistry,  geology,  and  anatomy.  . 

It  was  further  directed  that  each  annual  course  should 
be  made  the  basis  of  a  volume  to  form  part  of  a  series 
constituting  a  memorial  to  Mrs.  Silliman.  The  memo- 
rial fund  came  into  the  possession  of  the  Corporation 
of  Yale  University  in  the  year  1902;  and  the  present 
volume  constitutes  the  fourth  of  the  series  of  memorial 
lectures. 


155304 


PEEPACE 

IN  the  following  Lectures  which  were  delivered 
at  Yale  University,  October  22d  to  November  2d, 
1906,  I  have  given,  after  a  general  theoretical  in- 
troduction, a  resume  of  the  experimental  investi- 
gations which  I  have  carried  out  in  recent  years, 
with  the  aid  of  my  students,  on  chemical  equilibria 
at  high  temperatures. 

The  study  of  the  results  thus  far  obtained  in 
this  field  makes  it  appear  probable  that  there 
prevails  here  more  conformity  to  general  laws 
than  the  two  laws  of  thermodynamics  would  lead 
us  to  expect.  To  explain  these  regularities  I 
have  developed  a  new  theorem  which  seems  to 
reveal  new  truths  concerning  the  relation  between 
chemical  energy  and  heat.  It  can  hardly  be 
doubted  that  this  theorem  will  prove  useful  in 
the  treatment  of  questions  other  than  purely 
chemical,  but  in  the  following  Lectures  I  have 
not  entered  into  this  phase  of  the  subject. 

As  to  the  theorem  itself,  I  should  like  to  add 


yiii  PREFACE 

the  following  general  remarks.  The  large  mass 
of  experimental  data  upon  which  the  theorem  has 
been  successfully  tested  will  probably  remove 
any  doubt  as  to  whether  the  formulas  developed 
by  its  aid  have  disclosed  new  laws  to  us.  To  de- 
cide the  question  whether  the  theorem  represents 
only  an  approximate  principle  or  an  exact  law  of 
nature  similar  to  the  first  and  second  laws  of 
thermodynamics  will,  however,  necessitate  many 
further  investigations.  From  a  practical  point  of 
view  this  question  is  of  minor  importance,  as  my 
formulas  are  sufficiently  accurate  for  many  pur- 
poses. From  a  theoretical  standpoint  it  is,  how- 
ever, of  the  greatest  importance,  for  the  reason 
that  a  more  exact  formulation  of  the  theorem  may 
possibly  be  found. 

In  the  preparation  of  these  lectures,  and  in  the 
correction  of  the  proofs,  I  have  been  assisted  by 
Dr.  K.  George  Falk,  for  whose  willing  and  efficient 
services  I  wish  to  express  my  best  thanks. 

W.  N. 


CONTENTS 

LECTURE  I 

PAOK 

THE  GENERAL  APPLICATION  OF  THERMODYNAMICS  TO 

CHEMISTRY .  •      .        1 

LECTURE   II 

DERIVATION   OF   THE   EQUATION   OF    THE    REACTION 

ISOCHORE 10 

LECTURE  III 

NEW    EXPERIMENTAL     RESEARCHES     ON     CHEMICAL 

EQUILIBRIA  AT  HIGH  TEMPERATURES    ...      20 

LECTURE  IV 

NEW    EXPERIMENTAL     RESEARCHES     ON     CHEMICAL 

EQUILIBRIA  AT  HIGH  TEMPERATURES  (Concluded)      30 

j 
LECTURE  V 

INTEGRATION  OF  THE  EQUATION  OF  THE  REACTION 
ISOCHORE,  PRELIMINARY  DISCUSSION  OF  THE  UN- 
DETERMINED CONSTANT  OF  INTEGRATION,  AND  OF 
THE  RELATION  BETWEEN  THE  TOTAL  AND  THE 
FREE  ENERGIES  AT  VERY  Low  TEMPERATURES  .  39 


X  CONTENTS 

LECTURE   VI 

PAGE 

DETERMINATION  AND  EVALUATION  OF  THE  CONSTANT 
OF  INTEGRATION  BY  MEANS  OF  THE  CURVE  OF 
VAPOR  PRESSURE 53 

LECTURE   VII 

DETERMINATION  AND  EVALUATION  OF  THE  CONSTANT 
OF  INTEGRATION  BY  MEANS  OF  THE  CURVE  OF 
VAPOR  PRESSURE  (Concluded)  .  .  .68 

LECTURE   VIII 

THE  CALCULATION  OF  CHEMICAL  EQUILIBRIA  IN  HO- 
MOGENEOUS GASEOUS  SYSTEMS  .  .  .  .77 

LECTURE    IX 

THE  CALCULATION  OF  CHEMICAL  EQUILIBRIA  IN  HO- 
MOGENEOUS GASEOUS  SYSTEMS  (Concluded)  .  .  88 

LECTURE  X 

THE  CALCULATION  OF  CHEMICAL  EQUILIBRIA  IN  HET- 
EROGENEOUS SYSTEMS  AND  OF  ELECTROMOTIVE 
FORCES .96 


OF  THE 

UNIVERSITY 

OF 


LECTURE  I 

THE   GENERAL  APPLICATION  OF  THERMO- 
DYNAMICS TO  CHEMISTRY 

THE  application  of  the  first  law  of  thermody- 
namics to  chemistry  developed  thermochemistry, 
as  is  well  known. 

Let  us  consider,  for  example,  the  reaction  be- 
tween hydrogen  and  oxygen  in  the  formation  of 
water.  The  equation  is 

2H2  +  6>2  =  ZH2O  +  115300, 
in  which  115300  denotes  the  number  of  gram  calo- 
ries developed  in  the  production,  at  constant  vol- 
ume, of  two  gram  molecules  or  mols  of  water  in 
the  form  of  vapor  at  the  temperature  of  100°  C. 

If  the  reaction  is  allowed  to  take  place  at  con- 
stant pressure,  external  work  will  TDC  done,  when, 
as  in  our  example,  the  volume  is  changed  by  the 
reaction.  Such  changes  in  volume  can  generally 
be  disregarded,  except  at  exceedingly  great  press- 
ures, when  dealing  with  solids  or  liquids,  but  may 
be  considerable  in  the  case  of  gaseous  systems,  for 


2  THERMODYNAMICS  AND  CHEMISTRY 

which  they  can  be  calculated  from  the  gas  laws. 
For  every  additional  mol  of  gas  formed  in  a  reac- 
tion this  external  work  would  be  R  T,  where  R  is 
the  gas  constant  and  equal  to  1.985  if  we  take  the 
gram  calorie  as  the  unit  of  energy.  Therefore 
115300  +  1.985  X  373  would  be  the  heat  of  for 
mation  of  two  gram  molecules  of  water  vapor  at 
constant  pressure. 

In  its  most  general  form  the  equation  of  a  reac- 
tion may  be  written 

n&t  +  n2a2  -\  -----  +  ^1A1  +  v2A2  -\  ----- 
-  n/o/  +  ..-.+  vJAJ  +  vJAJ  +  •  •  •  • 

in  which  the  molecules  a  are  those  which  take  part 
in  the  solid  or  liquid  state,  and  the  molecules  A 
are  in  the  gaseous  state.  The  external  work 
would  be 

cal. 


The  heat  Q  of  a  reaction  at  constant  volume  can 
also  be  called  the  change  in  the  total  energy  taking 
place  during  the  reaction.  The  fundamental  prin- 
ciple of  thermochemistry,  derived  from  the  first 
law  of  thermodynamics,  is  that  Q  is  independent 
of  the  way  in  which  the  system  is  transferred  from 
one  state  to  the  other.  (Law  of  Constant  Heat 
Summation.) 


GENERAL  APPLICATION  3 

Q'y  the  heat  developed  at  constant  pressure,  can 
be  found  by  means  of  the  formula 
Q'=  Q  +  2vfiT. 

The  second  law  of  thermodynamics  states  that 
for  every  chemical  reaction  a  quantity  A,  the 
maximum  amount  of  work  which  can  be  obtained 
by  the  reaction  in  question,  also  called  the  change 
in  free  energy,  has  a  definite  value,  and  that  this 
maximum  work  will  be  done  if  the  reaction  pro- 
ceeds in  an  isothermal  and  reversible  manner. 
Moreover  the  value  of  A  is  absolutely  indepen- 
dent, like  Q,  of  the  way  in  which  the  system  is 
transferred  from  the  initial  to  the  final  state. 

The  value  of  A  —  Q,  the  excess  of  the  maximum 
work  of  an  isothermal  process  over  the  decrease  in 
total  energy,  denotes  the  quantity  of  heat  absorbed 
when  the  reaction  proceeds  in  an  isothermal  and 
reversible  manner,  and  is  called  the  latent  heat  of, 
the  reaction.  As  the  most  important  application  of 
the  second  law  of  thermodynamics  we  obtain  the 
following  expression  for  the  latent  heat  of  a  reac- 


(1)  A-Q= 

This  equation  contains  in  a  general  manner  all 
that  the  laws  of  thermodynamics  can  teach  con- 


4  THERMODYNAMICS  AND  CHEMISTRY 

cerning  chemical  processes.  A  and  Q  are  both 
expressed  in  units  of  energy  —  for  example,  gram 
calories  —  and  are  both  independent  of  the  way  in 
which  the  reaction  proceeds  under  the  aforesaid 
conditions. 

We  may  at  the  outset  emphasize  the  fact  that 
every  future  development  of  thermodynamics  will 
be  an  addition  to  the  above  equation.  As  a  matter 
of  history  it  may  be  stated  that  this  equation  was 
included  in  the  first  application  of  thermodynamics 
to  chemistry  by  Horstmann  (1869).  Shortly  after- 
ward the  problem  was  treated  very  thoroughly 
by  J.  Willard  Gibbs  in  his  great  work.  Later  the 
simplicity  and  clearness  of  this  formula  was 
pointed  out  by  Helmholtz. 

Let  us  consider  here  two  important  applications 
of  the  fundamental  formula. 

1.  The  physico-chemical  processes  such  as  vola- 
tilization, melting,  transformation  of  allotropic 
forms  into  each  other,  are  to  be  treated  in  exactly 
the  same  way  as  the  chemical.  For  example,  the 
well-known  formula  of  Clausius-Clapeyron, 

(2)  X= 


in  which  X  indicates  the  molecular  latent  heat  of 
vaporization,  p  the  vapor  pressure,  v  the  molecular 


GENERAL  APPLICATION  5 

«, 

volume  of  the  vapor,  and  vf  that  of  the  liquid,  can 
be  considered  as  a  direct  application  of  our  funda- 
mental formula.  If  the  vapor  pressure  is  not  too 
great,  vf  can  be  disregarded  in  comparison  to  v,  and 
the  equation  of  the  gas  laws, 

(3)  pv  =  Ml 
may  be  applied.     We  thus  obtain 

(4)  X 


The  gas  laws  can  also  be  written 

(5)  p  =  CRT, 

in  which  C  denotes  the  concentration  of  the  gas  or 
vapor.  Substituting  this  value  of  p  we  obtain 

(6)  \-RT=MT*^^-. 

dT 

In  this  equation  \—RT  corresponds  to  $,  being 
the  change  in  the  total  energy  connected  with  the 
process  of  condensation.  As  measurements  show, 
the  heat  of  vaporization  varies  in  a  continuous  and 
gradual  manner  and  can  therefore  be  formulated 

(7)  X  -  RT=  Xo  +  aT+  bT2  +  cT*  +  >... 

It  must  be  carefully  noted  that  this  equation 
while  applicable  to  every  substance,  liquid  or  solid, 
must  be  restricted  to  a  definite  form  of  the  sub- 
stance in  any  one  case.  We  are  therefore  justified 


6  THERMODYNAMICS  AND  CHEMISTRY 

in  applying  this  equation  to  liquid  water,  but  we 
must  suppose  the  water  to  be  undercooled  when 
we  apply  the  equation  to  very  low  temperatures. 
Also,  if  a  substance  can  exist  in  more  than  one 
crystallized  form,  the  above  formula  must  be  lim- 
ited to  one  definite  form  at  a  time.  There  are, 
therefore,  as  many  equations  of  the  above  form, 
but  with  different  numerical  values  of  the  coeffi- 
cients X0,  #,  b,  etc.,  as  there  are  different  condensed 
forms  of  the  substance. 

Integrating,  we  find  as  the  formula  for  the  con- 
centration of  the  saturated  vapor 


in  which  i  must  for  the  present  be  supposed  to  be 
characteristic,  not  only  for  any  one  substance,  but 
also  for  every  definite  form  of  that  substance. 

2.  It  is  interesting  to  note  that  our  fundamental 
formula  can  be  used  for  a  very  simple  classification 
of  all  natural  processes,  as  it  is  by  no  means  limited 
to  chemical  processes. 

Natural  changes  have  long  been  grouped  into 
physical  and  chemical.  In  the  former  the  compo- 
sition of  matter  usually  plays  an  unimportant  part, 
whereas  in  the  latter  it  is  the  chief  object  of  con- 


GENERAL  APPLICATION  7 

sideration.  From  the  point  of  view  of  the  molec- 
ular theory  a  physical  process  is  one  in  which  the 
molecules  remain  intact,  a  chemical  process  one  in 
which  their  composition  is  altered.  This  classifi- 
cation has  real  value,  as  is  shown  by  the  customary 
separation  of  physics  and  chemistry,  not  only  in 
teaching,  but  also  in  methods  of  research,  —  a  fact 
that  is  all  the  more  striking  as  both  sciences  deal 
with  the  same  fundamental  problem,  that  of  reduc- 
ing to  the  simplest  rules  the  complicated  phenom- 
ena of  the  external  world.  But  this  separation 
is  not  altogether  advantageous,  and  is  especially 
embarrassing  in  exploring  the  boundary  region 
where  physicists  and  chemists  need  to  work  in 
concert. 

Since  thermodynamic  laws  are  applicable  to  all 
the  phenomena  of  the  external  world,  a  classifica- 
tion based  upon  these  laws  suggests  itself.  The 
fundamental  formula 


involves  the  following  special  cases  : 

(a)  A  =  Q  :  the  changes  in  free  and  total  energy 
are  equal  at  all  temperatures.  Then  the  tempera- 
ture coefficient  of  A,  and  therefore  that  of  Q  also, 
is  zero,  that  is,  temperature  does  not  influence  the 


8  THERMODYNAMICS  AND  CHEMISTRY 

phenomenon  in  question,  at  least  not  as  regards  its 
tbermodynamic  properties.  Conversely,  if  the  last 
condition  is  fulfilled,  A  —  Q.  This  behavior  is 
shown  by  all  systems  in  which  only  gravitational, 
electric,  and  magnetic  forces  act.  These  can  be 
described  by  means  of  a  function  (the  potential) 
which  is  independent  of  temperature. 


(b)   Q  =  0,    and    therefore    A  =  T;   or 


A  =  a'l]  in  which  a!  is  a  constant  of  integration. 
A  is  then  proportional  to  the  absolute  temperature. 
The  expansion  of  a  perfect  gas  and  the  mixture  of 
dilute  solutions  are  the  instances  of  this  behavior 
in  which  the  influence  of  temperature  comes  out 
the  most  clearly.  (Gas  thermometer.) 

d  A 

(c)  A=Q,  and  therefore   Q  =  -  T-  -. 

dT 

This  condition  can  only  occur  at  single  points  of 
temperature;  but  A  can  be  small  in  comparison 
with  Q  over  a  considerable  range  of  temperature. 
As  then  the  percentage  variation  of  A  will  be 
large,  the  influence  of  temperature  must  be  very 
marked  in  such  cases  (evaporation,  fusion,  dissoci- 
ation, i.  e.,  all  properly  "physico-chemical"  phe- 
nomena). 

Case  (c)  is  evidently  not  so  simple,  and  has  not 


GENERAL  APPLICATION  9 

given  rise  to  such  important  hypotheses  as  case 
(a),  which  introduced  forces  of  attraction  into 
science,  and  case  (5),  which  was  decisive  in  the 
development  of  the  molecular  theory.  (Avogad- 
ro's  rule.) 

The  case  -4  =  0  and  Q  =  0  would  not  be  a 
process  in  the  thermodynamic  sense.  Such  cases, 
however,  exist  and  are  of  importance  (the  move- 
ment of  a  mass  at  right  angles  to  the  direction  of 
gravity,  passage  of  one  optical  isomer  into  the 
other,  etc.)  ;  so  it  appears  that  though  the  science 
of  thermodynamics  furnishes  important  points  of 
view  for  the  classification  of  phenomena,  it  is  too 
narrow  to  cover  the  whole. 

This,  of  course,  is  due  to  the  fact  that  the  two 
laws  of  thermodynamics  are  insufficient  for  a  gen- 
eral explanation  of  nature.  They  take,  for  ex- 
ample, no  account  of  the  course  of  phenomena  in 
time, — unlike  the  molecular  theory,  in  which  such 
a  limitation  has  not  thus  far  been  shown  to  exist. 


LECTURE  II 

DERIVATION     OF     THE     EQUATION     OF     THE 
EEACTION  ISOCHORE      = 


IN  the  following  discussion,  the  influence  of  the 
temperature  upon  the  heat  of  reaction  is  of  the 
greatest  importance.  The  law  of  the  conserva- 
tion of  energy  enables  us  to  calculate  this  influ- 
ence from  the  specific  heats  of  the  reacting  sub- 
stances. 

If  we  allow  the  same  reaction  to  occur,  once  at 
the  temperature  T  and  again  at  the  temperature 
T  -\-  dTj  the  heat  of  reaction  will  be  different  in 
the  two  cases  ;  let  these  heats  be  Q  and  Q  +  dQ 
respectively.  We  can  now  imagine  the  following 
cyclic  process  to  be  carried  out.  Let  the  reaction 
occur  at  the  temperature  T,  thereby  developing 
the  quantity  of  heat  Q  ;  after  which  the  tempera- 
ture of  the  system  is  raised  to  T  +  dT,  which  will 
require  the  introduction  of  h'  dT  gram  calories  of 
heat,  where  h'  denotes  the  heat  capacity  of  the 
substances  resulting  from  the  reaction.  Now  let 


REACTION  ISOCHORE  11 

the  reaction  occur  in  the  opposite  sense  at  T  -\-  dT, 
a  process  which  will  absorb  the  quantity  of  heat 
Q  +  dQ  ;  then  let  the  system  be  cooled  to  T, 
whereby  the  quantity  of  heat  Ti  dT  will  be  given 
off,  where  h  denotes  the  heat  capacity  of  the  re- 
acting substances.  The  system  has  now  returned 
to  its  original  condition. 

Now  the  law  of  conservation  of  energy  requires 
that  the  amount  of  heat  absorbed  by  the  system 
shall  be  the  same  as  that  given  out  ;  i.  e.,  that 

Q  +  dQ  +  h'dT=  Q  +  hdT, 
and  hence 


that  is,  the  excess  of  the  heat  capacity  of  the  re- 
acting substances  over  the  heat  capacity  of  the 
resulting  substances,  gives  the  increase  of  the  heat 
of  reaction  per  degree  of  temperature  elevation. 

The  following  diagram  illustrates  the  derivation 
of  the  above  equation  in  a  slightly  different  way  : 


—hdT 


+  Q+dQ         (g'A]  (T+dT) 


(aAHT)  +Q 

FIG.  1. 


-tidT 


12  THERMODYNAMICS  AND  CHEMISTRY 

From  the  same  initial  state,  corresponding  to 
the  molecules  (#,  A)  and  the  temperature  T,  we 
may  pass  by  two  different  paths  to  the  same  final 
state  (a',  A')  and  T-\-  dT.  Placing  the  changes  in 
total  energy  for  the  two  different  paths  equal  to 
each  other,  we  find 

Q  -  h'dT=  -  hdT+  Q  +  dQ, 
the  above  equation. 

The  great  importance  of  this  equation  is  due  to 
the  fact  that  with  its  aid,  when  we  know  the 
specific  heats,  we  can  calculate  the  influence  of 
temperature  upon  the  heat  of  reaction  much  more 
accurately  than  it  could  be  obtained  by  the  direct 
measurement  of  the  heat  of  reaction  at  two  dif- 
ferent temperatures. 

If  we  apply  the  heat  equation  to  the  process  of 
vaporization,  in  which 

Q=\-£T=\0  +  aT+ 
we  find 

(10)  = 


in  which 

(11)  Hv=a 

denotes  the  molecular  heat  of  the  vapor  at  con- 
stant volume,  and 

(12)  A0  =  a0 


REACTION  ISOCHORE  13 

denotes  the  molecular  heat  of  the  condensed  form 
of  the  vapor.     We  obtain  therefore 

a  =  a  —  a0,    I  —  /3  —  £o,    etc., 
and  by  substituting  in  (8) 


If  we  apply  the  above  equation  to  a  chemical 
process  taking  place  in  a  homogeneous  gaseous 
phase  with  coexistent  liquid  or  solid  substances, 
and  if  we  let 


we  find 

(14) 

Furthermore,  if  we  put 


we  obtain 
(16)  = 


and  therefore 

a  =  Sj'a  + 
(16a)  )b  =  2^/8  + 

etc. 


14  THERMODYNAMICS  AND  CHEMISTRY 

We  are  justified  in  assuming  the  above  relation 
to  hold  true  for  the  specific  heat  of  a  substance  at 
different  temperatures,  from  the  experimental  fact 
that  this  specific  heat  varies  continuously  and  only 
slightly  with  the  temperature. 

We  can  now  proceed  to  the  development  of  the 
formulas  of  chemical  equilibria  from  the  funda- 
mental formula 

s-\  m  (v^±- 


that  is  to  say,  to  the  derivation  of  the  equation 
of  the  reaction  isochore.  For  the  end  here  in  view 
I  desire  to  present  this  derivation  in  a  somewhat 
different  form  from  that  generally  followed. 

Let  us  consider  a  reaction  taking  place  between 
solid  and  liquid  substances  only,  that  is,  according 
to  our  notation  previously  adopted, 

^i  «i  +  n*a>2  H =  «  H ; 

or,  to  use  our  former  example,  the  formation  of 
two  mols  of  solid  or  liquid  water  from  two  mols 
of  solid  or  liquid  hydrogen  and  one  mol  of  solid 
or  liquid  oxygen. 

The  question  is:  How  can  the  formation  of 
water  under  these  circumstances  be  conceived  of 
as  taking  place  in  an  isothermal  and  reversible 


REACTION  ISOCHORE  15 

way?  For  this  purpose  two  mols  of  hydrogen 
and  one  mol  of  oxygen  are  vaporized  and  brought 
by  means  of  semipermeable  membranes  into  a 
space  which  we  may  call,  as  Haber  does,  the 
"  equilibrium  box "  and  in  which  H^  6>2,  and 
H^O  may  coexist  in  the  gaseous  state  in  equilib- 
rium. At  the  same  time  two  mols  of  the  water 
formed  may  be  supposed  to  be  removed  from  the 
equilibrium  box  through  a  suitable  semipermeable 
membrane  and  condensed  to  solid  or  liquid  water. 
The  concentrations  of  H%,  O2,  and  HZO  in  the 
equilibrium  box  may  be  represented  respectively 
by  <?!,  <?2,  and  <f. 

The  work  involved  in  transferring  one  mol  of 
gas,  formed  by  vaporization  under  the  vapor 
pressure  P,  in  a  space  in  which  the  partial  pres- 
sure of  the  gas  is  p,  is  given  by  the  expression 

PV+JSTfo—-pv, 

in  which  V  and  v  represent  the  volumes  of  the 
gas  or  vapor  under  the  corresponding  pressures  P 
and  p.  But  since  P  V=pv,  this  becomes  simply 

RTln-  =  RTlm,-, 

p  c ' 

in  which  C  and  c  stand  for  the  concentrations 


16  THERMODYNAMICS  AND  CHEMISTRY 

under  the  corresponding  circumstances.     In  con- 
veying n  mols  the  work  would  be,  of  course, 


G  C 

Now  we  can  easily  calculate  the  maximum  work 
done  when  hydrogen  and  oxygen  are  converted 
isothermally  and  reversibly  into  water,  all  the 
substances  being  in  the  solid  or  liquid  state.  We 
find 

A  = 


Oly  62,  <7',  being  respectively  the  concentrations  of 
the  saturated  vapor  of  hydrogen,  oxygen,  and 
water.  Transformed,  the  equation  becomes 

(17)       A 


The  maximum  work  A  must  be  independent  of 
the  nature  of  the  equilibrium  box,  which  only 
plays  the  part  of  an  intermediary  and  suffers  no 
change  during  the  process.  This  is  only  possible 
if  at  constant  temperature  the  expression 


c 
is  constant,  and  therefore, 


(18)  JT=  a- 

(JT  being  constant  at  constant  temperature). 


REACTION   ISOCHORE  17 

This  is,  however,  nothing  more  than  the  law  of 
mass  action,  applied  to  a  homogeneous  gaseous 
system. 

If  we  now  apply  the  fundamental  equation 


we  obtain 

-  Q 


ws,  ,  dlnC*      <bdlnC'\ 
~~        dT  dT     ) 


dT 
or 


dT 

\   dT     +   dT~         dT   )'' 
but  by  equation  (6) 


and  therefore 

-ST)  +  (\3 


-  2(X'  - 


, 


18  THERMODYNAMICS  AND  CHEMISTRY 

in  which  Xu  X2,  X'  correspond  respectively  to  HZJ 
O2,  H*>0.     The  expression  on  the  left  is  nothing 
but  the  heat  $',  developed  by  the  formation  of 
water  in  a  homogeneous  gaseous  system. 
This  gives 

(19)  qf 


the  equation  of  the  reaction  isochore. 

In  exactly  the  same  manner  we  find  in  the  most 
general  case  for  the  reaction 


the  corresponding  equations 

(20)  K=c 


It  may  further  be  added  that  substances  in  con- 
densed forms  also  can  exist  in  the  equilibrium 
box,  and  since  no  work  is  necessary  to  introduce 
or  remove  such  forms,  they  may  be  disregarded 
in  calculating  A.  This  is  simply  stating  the 
well-known  fact  that  the  active  mass  of  a  con- 
densed form  is  constant  (Guldberg). 

Finally,  for  a  gaseous  system,  if  instead  of  the 
solid  or  liquid  forms  we  take  the  substances  in 
the  gaseous  form  contained  in  large  reservoirs,  we 
obtain 


REACTION  ISOCHORE  19 

(21)  A  =  -RTlmK+  RTln  °0^  C«J*  "  ' ' 

Co/"1' 

in  which  the  values  of  <70  denote  the  concentra- 
tions of  the  gases  in  our  reservoirs.* 

Although  it  is  not  important  for  the  purposes 
here  in  view,  it  may  be  remembered  that  accord- 
ing to  the  theory  of  van't  Hoff  these  formulas 
hold  good  not  only  for  gaseous  systems  but  also 
for  dilute  solutions. 

*For  a  further  discussion  of  these  relations  consult  Planck's  "  Ther- 
modynaraik  "  and  the  author's  "  Theoretical  Chemistry." 


LECTURE  III 

NEW  EXPERIMENTAL  RESEARCHES  ON  CHEM- 
ICAL EQUILIBRIA  AT  HIGH  TEMPERATURES 

THE  formulas  which  we  deduced  in  the  last 
lecture  have  furnished  the  guiding  principles  for 
many  experimental  researches  concerning  chemical 
equilibria  which  have  been  carried  on  in  recent 
times.  In  the  hope  of  penetrating  more  deeply 
into  the  relations  between  chemical  energy  and 
heat,  I  have  carried  out  in  the  last  few  years, 
together  with  my  students,  a  number  of  investiga- 
tions on  reactions  at  high  temperatures  in  gaseous 
systems,  and  I  may  perhaps  be  allowed  to  give 
here  a  brief  account  of  this  work. 

I  began  by  extending  our  methods  for  the  de- 
termination of  molecular  weights  to  higher  tem- 
peratures, using  iridium  vessels  in  the  vapor  den- 
sity method  of  Victor  Meyer.  Heraeus  in  Hanau 
has  been  successful  in  recent  years  in  working 
this  material  in  a  very  skilful  manner.  For  heat- 
ing purposes  I  constructed  an  electric  furnace, 


NEW  EXPERIMENTAL  RESEARCHES  21 

also  of  iridium.  The  limitations  imposed  by  the 
great  cost  of  iridium  caused  quite  a  number  of 
changes  to  be  made  in  the  method  used  hereto- 
fore. 

The  details  of  the  apparatus*  can  be  seen  in 
Figs.  2  and  3.  Wide  copper  strips  were  fused 
to  the  iridium  tube  (furnace),  the  heating  being 
effected  by  a  current  having  an  energy  of  2000 
to  3000  watts.  By  this  means  a  temperature  of 
2000°  could  be  attained  and  kept  constant  for  a 
length  of  time  sufficient  to  carry  out  a  number  of 
successive  determinations.  The  tube  was  packed 
in  burnt  magnesia,  which  was  in  turn  enclosed  in 
an  asbestos  mantle,  leaving  access  to  the  two 
ends.  The  lower  end  of  the  tube  remained  open 
to  permit  the  temperature  of  the  inner  bulb  to  be 
determined  by  comparing  the  light  radiated  from 
it  with  the  intensity  of  the  radiation  from  a 
standardized  luminous  glower.  The  inner  bulb, 
in  which  the  substance  under  examination  was 
vaporized,  had  the  form  of  the  usual  Victor  Meyer 
apparatus,  but  necessarily  of  much  smaller  dimen- 
sions. The  upper  part  of  the  bulb  was  sur- 
rounded by  a  copper  spiral  through  which  water 
was  made  to  circulate,  and  the  substance  was 

*  Nernst,  Ztschr.  f.  Elektrochemie,  1903,  622. 


22 


THERMODYNAMICS  AND  CHEMISTRY 


retained  in  position  there  by  means  of  the  usual 
dropping  device,  till  the  bulb  was  heated  to 
the  required  temperature.  The  substances  were 
weighed  and  introduced  into  the  bulb  in  small 


FIG.  2. 

aa,  Magnesia   packing;  bb,  asbestos   mantle;    cc,  electrodes;  dd, 
wooden  support  (attached  to  wall  not  shown  in  the  figure). 

iridium  vessels.  The  weighings,  which  were  ac- 
curate to  0.001  to  0.002  mg.  were  made  on  a  sen- 
sitive micro-balance,  which  I  have  described  else- 
where. The  volume  of  air  displaced  by  the 
vaporization  of  the  substance  in  the  bulb  was 
indicated  by  the  motion  of  a  drop  of  mercury  in 


NEW  EXPERIMENTAL  RESEARCHES 


23 


a  glass  capillary  connected  to  the  bulb  by  a  piece 
of  rubber  tubing.  This  capillary  was  carefully 
calibrated,  and  the  increase  in  volume  could  thus 
be  determined  with  great  accuracy. 


o 

Iridium  Vessel 


n 


FIG.  3. 
(One  third  actual  size.) 


Of  the  results  obtained  by  the  method  described 
above,  the  following  may  be  mentioned:  The 
molecular  weights  of  H^O,  C02,  KOI,  NaCl,  S02, 
were  normal  at  temperatures  of  nearly  2000°; 


24  THERMODYNAMICS  AND  CHEMISTRY 

sulphur  was  almost  fifty  per  cent  dissociated  into 
atoms.  Silver  proved  to  be  monatomic,  as  was  to 
have  been  expected.* 

For  an  exact  determination  of  the  degree  of 
dissociation,  and  of  chemical  equilibria  in  general, 
the  above  method  is  unsuitable  because  the  par- 
tial pressures  in  the  Victor  Meyer  method  cannot 
be  accurately  measured,  f  For  this  purpose 
a  development  of  the  "streaming  method," 
first  used  by  Deville,  was  found  to  be  most  suit- 
able. 

The  mode  of  carrying  out  these  experiments 
may  be  made  clear  by  the  accompanying  sketch 
(Fig-  4)4 


FIG.  4. 

The  gaseous  mixture  to  be  studied  is  allowed 
to  flow  through  a  long  tube.  Between  the  points 
a  and  b  the  temperature  t,  at  which  the  equilibri- 
um is  to  be  investigated,  is  maintained,  while  from 

*  Wartenberg,  Berichte  d.  Deutsch.  Chem.  Gesell.,  39,  381. 
f  See,  however,  Loewenstein,  Ztschr.  f.  phys.  Chem.,  54,  711,  who 
used  a  very  ingenious  modification. 

J  Cf.  W.  Nernst,  Ztschr.  f.  anorg.  Chem.,  49,  213. 


NEW  EXPERIMENTAL  RESEARCHES      25 

b  to  c  the  temperature  is  made  to  fall  as  rapidly 
as  possible,  so  that  at  c  it  has  attained  such  a  low 
value  V  that  the  reaction  velocity  is  practically 
zero.  Evidently  the  following  two  conditions 
must  be  fulfilled  in  order  that  the  gas  leaving  the 
tube  shall  have  the  same  composition  as  at  the 
equilibrium  temperature :  first,  the  distance  ab 
must  be  sufficiently  long  to  allow  equilibrium  to 
be  attained ;  and  secondly,  the  cooling  space  be 
must  be  short  enough  not  to  change  this  equi- 
librium. 

The  first  condition  is  fulfilled  theoretically  by 
making  ab  sufficiently  long;  practically  this  can 
best  be  done  by  widening  the  tube  between  a 
and  b.  In  some  cases  a  catalytic  agent  will 
answer  the  same  purpose,  the  well-known  in- 
vestigations of  Knietsch  on  the  formation  of 
sulphur  trioxide  being  a  good  example  of  this. 
The  question  whether  the  reaction  velocity  is  suffi- 
ciently large  for  the  purpose  at  the  temperature  £, 
can  be  determined  by  passing  through  the  tube 
mixtures  whose  composition  is  made  to  lie  first  on 
one  side  and  then  on  the  other  of  the  composition 
of  the  equilibrium  mixture. 

The  second  condition  is  best  fulfilled  by  making 
be  a  narrow  capillary  in  order  to  give  as  great 


26  THERMODYNAMICS  AND  CHEMISTRY 

a  velocity  as  possible  to  the  gaseous  mixture,  and 
to  produce  as  large  a  fall  in  temperature  as  pos- 
sible. It  is,  however,  impossible  to  go  beyond  a 
certain  limit  here  on  account  of  the  conductivity 
for  heat  of  the  material  of  the  tube,  and  we  con- 
sequently cannot  conclude  that  this  source  of 
error  has  necessarily  been  avoided  if  the  composi- 
tion of  the  mixture  leaving  the  apparatus  is  inde- 
pendent of  the  speed  of  the  current  of  gas.  This 
follows  from  the  fact  that  an  infinitely  large 
velocity  of  the  gas  by  no  means  necessitates  an 
infinitely  rapid  fall  in  temperature.  Substances 
acting  catalytically  must  of  course  be  excluded 
from  be. 

Of  especial  importance  is  the  fact  that  at  high 
temperatures  and  correspondingly  great  reaction 
velocities,  equilibrium  is  certainly  reached  in  ab, 
but  is  just  as  certainly  disturbed  in  be.  The  gas 
leaving  the  apparatus  will  then  have  the  same 
composition,  no  matter  on  which  side  of  the  equi- 
librium the  composition  of  the  original  mixture 
lay,  but  in  spite  of  this  the  final  mixture  may 
differ  widely  from  the  true  composition  of  the 
equilibrium  mixture. 

The  accompanying  curves  (Fig.  5)  will  perhaps 
show  this  more  clearly;  the  unbroken  curve  is 


NEW  EXPERIMENTAL  RESEARCHES  27 

the  equilibrium  curve  (showing  percentages  of 
the  product  of  the  reaction),  while  the  dotted 
curve  represents  the  observed  values.  In  general, 
on  account  of  the  sources  of  error  mentioned,  the 
yield  obtained  will  be  too  small.  If,  however,  ab 
is  long  compared  to  be,  a  region  must  always  exist 
where  correct  values  are  obtained.  The  problem 
for  the  experimenter  is  to  find  the  temperature 


interval  T2  to  T3 ,  within  which  the  experimental 
values  are  correct,  and  which  will  evidently  extend 
farther  toward  the  left  the  greater  the  length  of  ab. 
A  very  important  control  in  locating  this  region 
of  correct  experimental  results  is  given  by  the 
fact  that  only  within  that  region  will  the  tangent 
of  the  observed  curve  coincide  with  that  of  the 
equilibrium  curve,  and  since  the  latter  can  be 
calculated  in  most  cases  from  the  heat  of  reaction, 


28  THERMODYNAMICS  AND  CHEMISTRY 

we  possess  a  trustworthy  criterion  as  to  whether 
the  observed  values  in  a  given  interval  agree  with 
the  true  values  for  the  equilibrium. 

In  cases  where,  as  in  the  neighborhood  of  T^ ,  the 
reaction  velocity,  although  perceptible,  is  still  too 
small  for  equilibrium  to  be  attained  in  ab  in  the 
given  time,  it  is  possible  by  passing  the  gas  mix- 
tures through  the  tube  at  different  rates  to  deter- 
mine the  two  opposite  reaction  velocities,  from 
which  the  concentration  of  the  mixture  at  equi- 
librium can  be  calculated  according  to  the  prin- 
ciple of  Guldberg  and  Waage,  as  was  first  pointed 
out  by  Dr.  J.  Sand,*  in  an  investigation  carried 
out  in  my  laboratory. 

Exactly  the  same  observations  apply  to  an  en- 
tirely different  kind  of  experiment.  In  the  case 
of  an  explosion,  the  gaseous  mixture  is  brought 
to  a  high  temperature,  remains  at  this  tempera- 
ture a  very  short  time,  and  is  then  rapidly  cooled. 
Since  very  small  intervals  of  time  are  here  in- 
volved, it  is  only  in  the  regions  of  great  reaction 
velocity  that  equilibrium  will  be  reached — that  is, 
at  very  high  temperatures,  where  the  method  de- 
scribed above  would  wholly  fail.  In  fact,  a  study 
of  the  explosion  of  mixtures  of  H%,  O2,  and  atmos- 

*  J.  Sand,  Ztschr.  f.  phys.  Chem.,  50,  465  (1904). 


NEW  EXPERIMENTAL  RESEARCHES      29 

pheric  air  has  proved  that  determinations  of  equi- 
libria can  be  made  at  temperatures  unattainable 
under  other  conditions.  It  is,  of  course,  necessary 
in  work  of  this  kind  to  prove  that  the  explosion 
temperature  lies  lower  than  Tz. 


LECTURE  IV 

NEW  EXPERIMENTAL  EESEARCHES  ON  CHEMI- 
CAL EQUILIBRIA  AT  HIGH  TEMPERATURES. 
— (Concluded). 

IN  the  last  lecture  it  was  pointed  out  that  the 
attainment  of  equilibrium  can  be  accelerated  by 
means  of  catalysts. 

The  case  is  especially  simple  if,  in  the  absence 
of  a  catalyst,  the  reaction  takes  place  at  only  a 
very  slow  rate.  It  would  then  evidently  suffice 
to  place  the  catalyst  in  the  space  with  the  gas  at 
the  desired  temperature,  and  to  analyze  the  gas 
after  a  short  time :  the  composition  of  the  result- 
ing mixture  would  correspond  to  the  equilibrium 
at  the  temperature  of  the  catalyst.  The  method 
is  very  simple  when  an  electrically  heated  con- 
ductor, for  example,  a  glowing  platinum  wire, 
has  a  sufficiently  strong  catalytic  action.  War- 
tenberg  and  I  observed  that  it  was  possible  to 
determine  the  dissociation  of  water  vapor  in  this 
way.  A  glowing  platinum  wire  in  water  vapor 
acts  in  such  a  way  that  after  a  time  the  water 


NEW  EXPERIMENTAL  RESEARCHES  31 

vapor  is  filled  with  the  products  of  dissociation 
to  an  extent  corresponding  to  the  temperature  of 
the  platinum  wire  and  the  pressure  of  the  water 
vapor.  This  method  was  worked  out  by  Lang- 
muir  *  in  a  recent  investigation,  in  which  the  dis- 
sociation equilibria  of  H2O  and  of  OO9  were 
determined  very  accurately. 

A  valuable  indication  of  the  reliability  of  the 
results  obtained  by  the  above  methods  may  be 
derived  from  the  law  of  mass  action,  by  carrying 
out  the  experiments  with  suitable  variations  of 
the  composition  of  the  mixtures. 

The  great  advantage  of  the  streaming  method 
described  above  consists  in  the  possibility  of 
simultaneously  determining  the  reaction  velocity. 
In  this  connection  the  work  of  my  assistant,  Jelli- 
nek,f  on  the  velocity  of  formation  and  of  decom- 
position of  nitric  oxide  (NO),  is  worthy  of  special 
mention. 

Finally,  a  very  ingenious  and  simple  method 
may  be  described,  which  was  discovered  and  used 
for  the  first  time  by  my  pupil  Loewenstein.J  In 
this  method  semipermeable  membranes,  whose 
theoretical  importance  was  illustrated  in  the  sec- 

*I.  Langmuir,  J.  Am.  Chem.,  Soc.  28,  1357. 
f  Jellinek,  Ztschr.  f.  anorg.  Chem.,  49,  229. 
\  Loewenstein,  Ztschr.  f.  phys.  Chem.,  54,  715. 


32 


THERMODYNAMICS  AND  CHEMISTRY 


ond  lecture,  were  put  to  practical  use.  Hydrogen, 
as  is  well  known,  diffuses  through  heated  plati- 
num and  palladium,  and  as  we  found,  also  through 
iridium  at  very  high  temperatures.  For  utilizing 
this  fact  the  following  apparatus  was  constructed. 
(Fig.  6.) 


Heated  Tube 


coup 


((KrS^ 

)le^    KsSST               I                            ^                                  ^ 
7/<*>    IP*                                                                         «*W 

Jl 

-o 

Pump 

I 

-i 

-2 

- 

-3 

" 

~4 

-;, 

/ 

-r, 

Fia.  6. 

A  platinum  bulb  8  cm.  long  and  1.2  cm.  in  ex- 
ternal diameter  was  connected  by  means  of  a 
capillary  tube  12  cm.  long  and  0.6  mm.  in  diam- 
ter  to  an  oil  manometer  and  mercury  air  pump. 
The  bulb  was  placed  in  the  center  of  the  hori- 
zontal tube,  which  was  heated  to  the  desired  tem- 
perature, and  water  vapor,  or  any  other  gas  which 
was  to  be  examined,  was  passed  through  the  tube. 
In  carrying  out  a  determination,  the  bulb  was 
first  evacuated,  the  oil  in  the  two  arms  of  the 
manometer  brought  to  the  same  level  by  opening 
the  stopcock  shown  in  the  figure  and  then  closing 


NEW  EXPERIMENTAL  RESEARCHES  33 

it,  and  finally  a  regular  stream  of  the  gas  whose 
dissociation  was  to  be  determined  was  passed 
through  the  tube  while  the  latter  was  heated  to  a 
constant  temperature.  Hydrogen  diffused  through 
the  walls  of  the  bulb,  and  the  vacuum  above  the 
arm  of  the  manometer  communicating  with  the 
pump  being  kept  constant,  the  difference  in  level 
of  the  oil  in  the  two  arms  gave  the  pressure  of 
the  hydrogen  which  had  entered  the  bulb.  After 
a  few  minutes  this  difference  became  constant  and 
equal  to  the  partial  pressure  of  the  hydrogen  in 
the  water  vapor  surrounding  the  bulb  in  the  tube, 
and  produced  by  the  dissociation  of  the  vapor. 
The  temperature  of  the  vapor  in  the  tube  was 
measured  with  the  aid  of  a  thermo-couple.  By 
this  method  the  dissociations  of  water  vapor,  hy- 
drochloric acid,  and  hydrogen  sulphide  were 
measured. 

In  these  experimental  researches,  which  cover  a 
period  of  about  eight  years,  various  forms  of  elec- 
tric resistance  furnaces  were  used.  Besides  those 
already  described,  two  other  forms  may  be  men- 
tioned. 

(1)  For  temperatures  up  to  1000°,  a  copper 
tube  coated  with  soapstone  and  water  glass  and 
wound  with  nickel  wire  allows  a  very  uniform 


34  THERMODYNAMICS  AND  CHEMISTRY 

temperature  to  be  maintained  within  the  tube, 
owing  to  the  good  conductivity  of  copper  for 
heat.* 

(2)  In  equilibria  in  which  carbon  takes  part,  an 
electric  furnace  of  carbon  permits  of  the  employ- 
ment of  very  high  temperatures.  In  this  way 
Rothmund  f  in  1901  determined  the  equilibrium  of 
the  formation  of  calcium  carbide. 

Iridium  when  exposed  to  the  air  at  high  tem- 
peratures becomes  disintegrated  and  crystalline 
on  the  surface,  probably  owing  to  the  formation 
of  a  volatile  oxide,  stable  only  at  high  tempera- 
tures. To  prevent  this  it  was  found  best  to  coat 
the  iridium  used  with  a  thin  layer  of  oxides  of 
zirconium  and  yttrium.  The  iridium  for  this 
purpose  is  painted  with  a  solution  of  eighty  per 
cent  zirconium  nitrate  and  twenty  per  cent  yttrium 
nitrate  and  then  heated  to  redness.  This  treat- 
ment is  repeated  twenty  to  thirty  times.  By 
quantitative  measurements  I  was  able  to  show 
that  the  loss  of  iridium  when  heated  in  air  to 
about  2000°  was  very  much  decreased  by  this 
treatment. 

For  temperatures  up  to  about  1650°  a  furnace 

*  Described  for  the  first  time  by  my  pupil  Hunter,  Ztschr.  f .  phys. 
Chem.,  53,  441. 

f  Rothmund,  Gottinger  Nachrichten,  1901,  Heft  3. 


NEW  EXPERIMENTAL  RESEARCHES  35 

constructed  of  platinum  exactly  similar  to  the 
iridium  furnace  described  is  very  durable  and 
useful.  A  furnace  of  platinum  with  about  twenty 
to  thirty  per  cent  iridium  is  available  up  to  1800°. 
In  the  following  tables,  I  and  II,  are  given  the 
results  obtained  for  the  technically  important  equi- 
libria of  the  reactions 


and 


TABLE  I. — FORMATION  OF  NITRIC  OXIDE 


T 

x  (obs.) 

x  (calc.) 

OBSERVER 

1811 

0.37 

0.35 

Nernst  (1.  c.). 

1877 

0.42 

0.43 

Nernst  and  Jellinek  (1.  c.). 

2023 

0.52  <...<  0.80 

0.64 

U              ((               U                       U 

2033 

0.64 

0.67 

Nernst  (1.  c.). 

2195 

0.97 

0.98 

U                  U 

2580 

2.05 

2.02 

Nernst  and  Finkh.* 

2675 

2.23 

2.35 

«(               U                 (( 

The  last  two  values  were  obtained  by  the  ex- 
plosion method,  the  others  by  the  streaming 
method.  In  the  table  x  is  the  percentage  by 
volume  of  NO  formed  in  atmospheric  air,  the  law 
of  mass  action  furnishing  therefore  the  relation 

/y,2 


I79'2"  I 

Nernst  and  Finkh,  Ztschr.  f.  anorg.  Chem.,  45  (1905),  116  and  126. 


36 


THERMODYNAMICS  AND  CHEMISTRY 


By  integrating  the  equation  of  the  reaction 
isochore,  placing  Q  equal  to  43200  independently 
of  the  temperature,  we  find 

43200 


4.571  T 


const. 


The  agreement  between  the  observed  and  cal- 
culated values  for  x  is  very  satisfactory,  as  shown 
in  the  foregoing  table,  if  we  let 

const.  =  2  [Iog10  0.0249  +  2.148]  =  1.09. 

TABLE  II. — DISSOCIATION  OF  WATER  VAPOR 


T 

x  (obs.) 

x  (calc.) 

OBSERVER 

1300 

0.0027 

0.0029 

Langmuir  (1.  c.). 

1397 

0.0078 

0.0084 

Nernst  and  Wartenberg.* 

1480 

0.0189 

0.0185 

u             u                  u 

1500 

0.0197 

0.0221 

Langmuir  (1.  c.). 

1561 

0.034 

0.0368 

Nernst  and  Wartenberg.  f 

2155 

1.18 

1.18 

Wartenberg.  f 

2257 

1.77 

1.79 

H 

The  results  of  Nernst  and  Wartenberg  were 
obtained  by  the  streaming  method,  those  of 
Wartenberg  by  the  use  of  "  semipermeable  mem- 
branes "  (iridium),  and  those  of  Langmuir  by  the 
catalytic  action  of  a  heated  platinum  wire. 

*  Nernst  and  Wartenberg,  Ztschr.  f.  phys.  Chem.,  56,  534. 
f  Wartenberg,  Ztschr.  f.  phys.  Chem.,  56,  513. 


NEW  EXPERIMENTAL  RESEARCHES  37 

Since  x  denotes  the  percentage  of  dissociation, 

we  have 

%Px  Px 

=  122X200  +  00'    C*~~~~  RT(ZW  +  xy 
2P(100  -  x) 


and  therefore 


RT       (200  +  x)  (100  -  x)2 

If  we  take  for  the  molecular  heats*  of   H2 

and  O2 

Hv  =4.68  +  0.000262; 

and  for  that  of  H%O 

Hv'  =  6.61  +  0.000717  T+  3.12-10-7T2, 
we  obtain 

#'=114400  +  2.74T— 0.00063T2  — 6.24-10-7T8 
and  hence,  by  integrating  the  equation  of  the  re- 
action isochore, 

25030 


,      _      _ 

log  -  =  11.46 

-4-    X  Wl  X  V 

"TooJv    "looj 


+  2.38  log  -—  -  1.38-  10-4  (T-  1000) 
D1000 

-0.685-10-7[T2-  (1000)8]. 

*  Wartenberg,  Verhand.  Deutsch.  Phys.  Gesell.,  8,  97  (1906). 


38  THERMODYNAMICS  AND  CHEMISTRY 

The  values  calculated  from  this  equation  (given 
in  the  table  under  x  (calc.))  show  a  very  satis- 
factory agreement  with  the  experimental  data. 

The  above  tables  prove  that  the  different 
methods  employed  give  results  which  are  in  com- 
plete thermodynamic  agreement,  the  differences 
being  no  greater  than  may  be  explained  by  the 
errors  in  the  measurement  of  the  temperatures. 
At  the  same  time  these  tables  represent  the  ap- 
plication of  thermodynamics  to  gaseous  systems 
at  temperatures  which  are  perhaps  the  highest  at 
which  such  investigations  have  been  carried  out 
up  to  the  present  time. 

In  a  similar  manner  a  number  of  other  reac- 
tions have  been  studied  in  my  laboratory,  among 
which  the  following  may  be  mentioned  : 


S,  +  0=  08,. 

*  Nernst  and  Wartenberg,  Ztschr.  f  .  phys.  Chem.,  56,  548. 
f  0.  F.  Tower,  J.  Am.  Chem.  Soc.,  27,  1209. 


LECTUEE  V 

INTEGRATION  OF  THE  EQUATION  OF  THE 
REACTION  ISOCHORE,  PRELIMINARY  DIS- 
CUSSION OF  THE  UNDETERMINED  CON- 
STANT OF  INTEGRATION,  AND  OF  THE 
RELATION  BETWEEN  THE  TOTAL  AND  THE 
FREE  ENERGIES  AT  VERY  LOW  TEMPERA- 
TURES. 

FOR  the  relation  between  chemical  energy  and 
heat  development  Berthelot,  as  is  well  known, 
believed  he  had  found  a  very  simple  expression 
when  he  set  the  two  magnitudes  equal  to  each 
other.  Closely  connected  with  this  question  is 
the  so-called  "Thomson's  Rule,"  proposed  at  an 
earlier  date,  according  to  which  the  electrical 
work  furnished  by  a  galvanic  cell 'is  equal  to  the 
heat  developed  by  the  chemical  reaction  producing 
the  current. 

We  have  seen,  however,  in  a  previous  lecture, 
that  there  are  formulas  by  means  of  which  the 
maximum  work  of  a  reaction  can  be  calculated 
with  the  aid  of  the  constant  K  of  the  law  of  mass 


40  THERMODYNAMICS  AND  CHEMISTRY 

action,  and  when  these  were  known  it  became 
possible  to  make  an  exact  quantitative  comparison 
between  A  and  Q  for  a  large  number  of  chemical 
reactions.  Another  method  was  found  in  the  cal- 
culation of  A  from  the  electromotive  forces  of 
such  galvanic  elements  as  are  reversible,  and 
therefore  available  for  this  purpose.* 

It  then  became  clear  that  the  maximum  work 
is  not  by  any  means  equal  to  the  heat  effect  de- 
termined thermochemically, — in  fact,  we  can  go 
a  step  further  and  say  that  it  is  often  wholly 
illogical  to  compare  these  two  quantities  directly 
with  each  other. 

The  heat  ($)  developed,  for  example,  in  the 
formation  of  water  vapor  from  gaseous  hydrogen 
>and  oxygen,  is  independent  of  the  concentrations 
of  the  reacting  substances,  but  the  maximum  work 
(A)  given  by  the  formula  already  derived 


(21)     A  =  -  KTtnK+  RTln    *  ^ 

depends  on  the  concentrations  of  both  the  react- 
ing substances  and  of  the  water  vapor  formed. 
We  can  therefore  let  A  assume  any  magnitude 
we  choose  by  suitably  varying  the  concentra- 

*  Cf.  Nernst,  "Theoretical   Chemistry"   (English  translation  of 
the  fourth  German  edition),  p.  685  ff.  and  p.  712  ff. 


RELATION  BETWEEN  TOTAL  AND  FREE  ENERGIES    41 

tions,  whereas  Q  always  retains  the  same  value. 
Further,  if  we  consider  our  fundamental  formula 


and  if,  according  to  the  principle  proposed  by 
Berthelot,  A  =  Q  at  all  temperatures,  then  both  A 
and  Q  must  be  independent  of  the  temperature. 
The  constancy  of  Q  would  require  the  existence 
of  certain  relations,  already  explained,  between 
the  specific  heats  of  the  substances  taking  part  in 
the  reaction,  but  experiment  has  shown  that  in 
general  these  conditions  are  not  fulfilled. 

Upon  attempting  to  find  the  mathematical  rela- 
tions between  A  and  Q  it  can  easily  be  seen  that 
A  cannot  be  calculated  from  Q  by  means  of  the 
two  laws  of  thermodynamics,  for  if 


were  a  solution  of  our  fundamental  formula,  then 

A  =f(T)  +  a'T, 

in  which  af  is  an  absolutely  indefinite  constant, 
would  be  a  solution  also. 

We  have  therefore  arrived  at  the  two  following 
results,  which  must  be  stated  before  any  further 
development  of  the  theory  can  be  attempted. 
(1)  The  relation 

A=  Q 


42  THERMODYNAMICS  AND  CHEMISTRY 

is  contrary  to  the  results  of  experiment,  and  is 
often  hardly  logical. 

(2)  The  principles  of  thermodynamics  do  not 
enable  us  to  find  the  relation  between  A  and  Q  ; 
i.  e.,  to  calculate  a  chemical  equilibrium  from  the 
heat  of  reaction. 

Moreover,  the  correctness  of  the  second  state- 
ment can  be  clearly  shown  by  integrating  the 
equation  of  the  reaction  isochore.  By  combining 
equations  (16)  and  (19)  we  find 

(22) 


. 
in  which  /  is  a  constant  of  integration  whose 

value  is  thus  far  entirely  undetermined.  There- 
fore if  a  new  law  of  thermodynamics  is  to  be 
found,  it  is  clear  from  the  outset  that  it  must  con- 
cern the  above  constant  of  integration  as  the  only 
remaining  problem. 

Can  we  hope  to  derive  such  a  law?  I  have 
thought  for  a  long  time  that  this  question  was  to 
be  answered  affirmatively.  In  the  different  edi- 
tions of  my  "Theoretical  Chemistry"  I  have 
stated  that  in  the  Principle  of  Berthelot,  even  if 
it  is  incorrect  in  the  form  used  up  to  the  present, 
there  lies  hidden  a  law  of  nature,  the  further 


RELATION  BETWEEN  TOTAL  AND  FREE  ENERGIES    43 

development  of  which  seems  to  be  of  the  greatest 
importance. 

To  enable  us  to  proceed  it  is  necessary  to  find 
the  conditions  under  which  the  Principle  of  Ber- 
thelot  comes  nearest  to  expressing  the  true  rela- 
tion between  chemical  energy  and  heat,  or,  what 
amounts  to  the  same  thing,  between  the  magni- 
tudes A  and  Q.  In  this  direction  we  can  show 
that  in  reactions  between  solids,  liquids,  or  con- 
centrated solutions  the  values  of  A  and  Q  ap- 
proach each  other  very  closely,  while  on  the  other 
hand,  in  dilute  solutions  or  with  gases  we  usually 
find  large  differences  between  the  two  quantities ; 
but  in  these  latter  cases,  as  we  have  seen  already, 
the  comparison  is  not  permissible  on  account  of 
the  nature  of  the  formulas. 

As  examples  illustrating  the  facts,  let  us  com- 
pare  the  electromotive  forces  of  some  galvanic 
cells  with  the  heats  developed  by  the  chemical 
reactions  taking  place  in  them. 
TABLE  A 


CHEMICAL  REACTION 

A 

Q 

2Hq  4-  PbCL  —  PI  4-  2HqCL. 

0.54 

0.44 

2Ag  +PbClz   —  Pb  +  2AgCl  

0.49 

0.52 

2Ag  -}-  /„  —  2  Agl  

0.68 

0.60 

Pb  _[-  72  —  Pblz  

0.89 

0.87 

44  THERMODYNAMICS  AND  CHEMISTRY 

In  the  table  A  and  Q  are  both  expressed  in  volts. 
Similarly,  the  electromotive  force  of  the  well- 
known  lead  storage  cell,  when  concentrated  sul- 
phuric acid  is  used,  is  almost  exactly  equal  to  the 
thermochemical  energy. 

Further,  Bodlander*  found  in  1898  that  he 
could  calculate  the  solubility  of  salts  from  their 
heats  of  formation  and  their  decomposition  poten- 
tials, and  he  pointed  out  very  clearly  that  the 
agreement  between  the  experimental  and  the  cal- 
culated solubilities  was  satisfactory  only  when  the 
decomposition  products  of  the  electrolysis  were 
solid  substances.  Silver  iodide  (Agl)  may  be 
mentioned  as  an  example. 

Furthermore,  I  found  in  1894f  in  comparing 
the  change  in  total  energy  with  that  of  free 
energy,  or  in  other  words  the  heat  of  reaction 
with  the  osmotic  work,  in  concentrated  solutions 
of  sulphuric  acid,  that  these  two  magnitudes  are 
nearly  equal  to  each  other,  whereas  in  dilute  solu- 
tions the  difference  is  very  great. 

But  even  in  the  cases  mentioned  above,  there  is 
no  doubt  that  the  principle  of  the  equality  of  A 
and  Q  at  ordinary  temperatures,  is  far  from  an 

*  Bodlander,  Ztschr.  f.  phys.  Chem.,  27,  55. 
t  Nernst,  Wiederaann's  Annalen,  53,  57. 


RELATION  BETWEEN  TOTAL  AND  FREE  ENERGIES    45 

exact  law.  Not  only  do  the  differences  between 
A  and  Q  exceed  the  errors  of  observation,  but  the 
consideration  of  the  physico-chemical  process  of 
fusion  also  proves  in  a  most  striking  manner  that 
A  and  Q  can  differ  very  greatly,  even  when  only 
solids  and  liquids  take  part  in  the  transformation. 
In  fact  at  the  melting  point,  A  is  almost  exactly 
zero,  whereas  Q,  the  latent  heat  of  fusion,  has  a 
considerable  value. 

A  long  study  of  this  relation  in  past  years  has 
led  me  to  the  hypothesis  that  we  are  dealing  with 
a  law  more  or  less  approximate  at  ordinary  tern- 
peratures,  but  true  in  the  neighborhood  of  absolute 
zero. 

That  A  and  Q  are  exactly  equal  at  the  absolute 
zero  is  a  necessary  consequence  of  the  fundamental 
formula 


when  Q,  and  therefore  also  A,  is  supposed  to  be  a 
continuous  function  of  Tdown  to  absolute  zero; 
but  what  I  should  like  to  point  out  is  that  A  and 
Q  are  not  only  equal  to  each  other  at  the  absolute 
zero,  but  also  that  their  values  coincide  completely 
in  the  immediate  vicinity  of  this  point.  To  illus- 
trate graphically,  the  curves  for  Q  and  A  (Fig.  7), 


46 


THERMODYNAMICS  AND  CHEMISTRY 


not  only  terminate  in  the  same  point  at  absolute 
zero  (cf.  I),  but  are  also  tangent  to  each  other 
(cf.  II). 

I  was  very  much  surprised,  in  following  up  the 
consequences  of  this  hypothesis,  to  find  that  it  con- 
tains the  solution  of  the  problem  concerning  the 


FIG.  7. 

relations  between  the  chemical  affinity  A,  and  Q 
the  heat  developed.  The  consideration  of  these 
consequences  and  the  calculation  of  practical  ex- 
amples will  form  the  subject  of  the  following  lec- 
tures. 

We  must  not  forget  that  we  are  here  dealing 
with  a  hypothesis  which  we  cannot  verify  directly, 


RELATION  BETWEEN  TOTAL  AND  FREE  ENERGIES    47 

because  we  are  unable  to  measure  Q  and  A  at  tem- 
peratures in  the  vicinity  of  the  absolute  zero.  It 
is  clear,  however,  that  a  knowledge  of  specific  heats 
down  to  absolute  zero  would  enable  us  to  under- 
take an  accurate  test  of  our  hypothesis,  but  even 
if  these  values  are  not  known,  we  can  in  many 
cases  extrapolate  with  sufficient  accuracy  to  fur- 
nish an  adequate  confirmation. 

To  apply  our  hypothesis,  let  us  consider  a  re- 
action 

n^i  +  n2a2  +  ....=  T&/O/  +  %'«/  -\  — 

between  solid  and  liquid  substances  only.     Ac- 
cording to  our  previous  assumption  we  set 
(22a)      Q=  Q0  +  aT+bT*  +  cT*  +  ..  .  ., 

in  which  the  coefficients  a,  5,  c,  .  .  .  .  are  to  be  cal- 
culated from  the  specific  heats. 

a=^na0l  b  =  2<n/3o,  etc. 

We  shall  attempt,  by  means  of  the  additional 
hypothesis*  that  A,  like  Q,  may  be  expressed  by  a 
series  containing  only  integral  pmvers  of  T,  to  de- 
termine whether 
(23)      A=A0  + 


*  Such  hypotheses  are  usual  in  thermodynamics  and  other  branches 
of  theoretical  physics,  but  it  does  not  seem  to  be  customary  to  point 
out  that  such  expansions  into  series  are  hypothetical  and  are  only  to 
be  justified  by  experimental  investigations. 


48  THERMODYNAMICS  AND  CHEMISTRY 

represents  a  correct  solution  of  our  fundamental 
formula 

A-Q=TdA 
We  find  by  substitution 


-\  -----  = 

and  since  the  last  equation  must  hold  at  all  tem- 
peratures, the  relations 


cf  —  c  =  3o',  etc., 

follow  as  necessary  and  sufficient  conditions,  from 

which  we  obtain 

(24)   A0=Q0,  a=0,  b  =  -b',  c=-2c',  etc. 

The  second  law  of  thermodynamics  requires 
then  (upon  the  above  very  probable  assumption 
that  both  A  and  Q  may  be  expressed  by  develop- 
ment into  series  containing  only  integral  powers 
of  T),  that  at  absolute  zero  A,  the  chemical 
affinity,  must  be  equal  to  Q,  the  heat  developed, 
and  further  that 

a  —  2^a0  =  0. 

That  is,  the  specific  heats  are  additive  at  absolute 
zero,  a  relation  which  has  been  found  to  be  ap- 
proximately true  for  solid  bodies  at  ordinary  tern- 


RELATION  BETWEEN  TOTAL  AND  FREE  ENERGIES    49 

peratures.  For  liquids  the  law  does  not  hold 
even  approximately  at  ordinary  temperatures,  and 
I  think  the  explanation  for  this  fact  is  simply 
that  the  specific  heats  of  liquids  increase  much 
more  rapidly  with  rise  in  temperature  than  those 
of  solids. 

The  essential  point  of  our  new  hypothesis 


(25)  fe.=  when  T= 


can  now  be  expressed  very  concisely  by  the  equa- 

tion 

(26)  a'  =  0. 

If  we  do  not  wish  to  make  the  hypothesis  that 
A  may  be  expressed  by  development  into  a  series 
containing  only  integral  powers  of  T,  it  would  be 
easy  to  integrate  equations  (1)  and  (22a),  obtaining 
thereby  the  following  equation  : 


From  the  hypothesis 


-alnT—2bT—....}  when  T=0, 

the  same  result  follows  as  before,  namely, 
a  =0  and  a'  =  0. 


50  THERMODYNAMICS  AND  CHEMISTRY 

The  simplest  application  of  the  above  formulas 
would  be  the  calculation  of  the  electromotive  force 
of  the  galvanic  cells  which  we  considered  on  p.  43, 
from  the  heats  of  reaction  and  the  specific  heats. 
In  fact  we  find  * 

A  =  7^23046  =  #o—  T2  2^/30  H  -----  , 

from  which  JEcan  be  calculated  in  volts  if  the  heats 
of  reaction  are  expressed  in  gram  calories  per  n 
electrochemical  equivalents.  But  since  the  ex- 
perimental data  are  not  sufficient  for  an  exact  test 
of  the  above  relation,  we  shall  consider  here  only 
the  simple  equilibrium 

a    —    a', 

(liquid)        (solid) 

that  is,  the  process  of  solidification. 

The  maximum  work  which  can  be  gained  by 
this  reaction  is 


where  O  and  G'  denote  the  concentrations  of  satu- 
rated vapor  at  the  temperature  T.  Our  theory 
requires  that 


*  With  regard  to  the  numerical  factor  in  this  equation,  cf.  the 
author's  "  Theoretical  Chemistry,"  p.  703,  and  also  Ztschr.  f.  Elektro- 
chemie,  10,  629  (1904). 


RELATION  BETWEEN  TOTAL  AND  FREE  ENERGIES  51 

If  the  molecular  heat  of  the  liquid  is 


and  that  of  the  solid 
V  =  o«+  2 
the  latent  heat  of  fusion  will  be 


The  melting  point  T0  is  determined  by  the  con- 
dition, 

A  =  o=    -     -  ~ 


or,  by  disregarding  the  higher  terms, 

7*2  _       vo 

"~  ' 


If  we  introduce  the  molecular  heats  at  the  melt- 
ing point, 

J5&=oo+2ft;ro  and  ^  =  00+2^^0, 
it  follows  that 
/oo\  T>  -         2^o       _        ^ 

(28)        ^-JI^HJ-HO-H^ 

in  which  cr  denotes  the  heat  of  fusion  at  the  melt- 
ing point  T0,  and  therefore 


Our  theory  allows  us,  therefore,  to  calculate  the 
melting  points  from  thermal  magnitudes  only,  and 


52  THERMODYNAMICS  AND  CHEMISTRY 

if  it  were  justifiable  to  assume  that  the  specific 
heats  for  the  liquid  and  solid  states  increase  line- 
arly with  the  temperature,  the  melting  points  could 
be  found  by  dividing  the  molecular  heat  of  fusion 
by  the  difference  between  the  molecular  heats  of 
the  solid  and  liquid  forms  at  the  melting  point. 
Indeed  Tammann  in  his  admirable  work,  "Krys- 
tallisieren  und  Schmelzen,"  Leipzig,  1903,  p.  42, 
showed,  from  the  empirical  side,  that  this  relation 
holds  true  in  many  cases.  Evidently  it  is  not 
always  permissible  to  disregard  the  higher  powers 
of  T.  It  would  be  almost  hopeless  to  attempt  a 
direct  experimental  verification,  because  the  un- 
dercooling of  the  liquid  below  the  solidification 
temperature  which  would  be  necessary  in  order 
to  determine  the  specific  heats  at  very  low  tem- 
peratures, would  be  impossible ;  but  perhaps  we 
may  hope  to  find  a  theoretical  method  which 
will  enable  us  to  determine  these  values. 

We  are  thus  able  to  prove  in  qualitative  agree- 
ment with  our  hypothesis,  that  the  specific  heat 
of  a  substance  is  always  greater  in  the  liquid 
than  in  the  solid  state ;  otherwise  an  equilibrium, 
that  is,  a  melting  point,  could  not  exist, — at  least 
not  at  the  pressure  of  the  saturated  vapor. 


LECTURE  VI 

DETERMINATION  AND  EVALUATION  OF  THE 
CONSTANT  OF  INTEGRATION  BY  MEANS 
OF  THE  CURVE  OF  VAPOR  PRESSURE 

AMONG  the  relations  deduced  in  the  second  lec- 
ture we  found,  for  a  reaction 


the  two  equations 

Q=QJ  + 

and 

A  =  RTln  CV°\  -  RTlnK. 


Substituting  in  the  latter  from  equation  (13) 
the  expression  for  the  concentrations  of  saturated 
vapor 


and  from  equation  (22)  the  expression  for  the 
equilibrium  of  a  homogeneous  gaseous  system 


54  THERMODYNAMICS  AND  CHEMISTRY 

we  find,  since        Q0  =  QQ'  — 


By  comparison  with  equation  (24)  we  obtain 
tf=2noo  —  0,   and   a'  =  2(^'  —  I)  It  =  0. 

The   constant   of  integration   I,  which  as  we 
have  seen  is  not  determinable  by  the  second  law 
of  thermodynamics,  is  therefore  given  according  to 
our  theory  by  the  equation 
(29)  1  =  2ni. 

That  is,  the  constant  I  is  referred  to  a  sum  of  con- 
stants of  integration  i  which  are  peculiar  to  each 
individual  substance,  and  can  be  found  by  separate 
measurements  carried  out  on  every  substance. 

We  have  found  further  that  by  equation  (24) 

b  =  -  V, 

and  also  that  5  =  -f-  ^n/30. 

Hence  ~bf  —  —  2^/30. 

It  is  historically  worthy  of  note  that  Boltzmann  * 
in  1882  pointed  out  that  a  kinetic  treatment  of 
gaseous  equilibrium  should  theoretically  lead  us 
farther  than  could  the  application  of  the  principles 
of  thermodynamics,  but  no  new  results  of  prac- 
tical value  have  as  yet  been  found  in  this  way. 

After  having  finished  the  above  calculations  I 

*  Boltzmann,  Wiedemann's  Annalen,  22,  64. 


DETERMINATION  OF  THE  INTEGRATION  CONSTANT    55 

found  in  the  monograph  of  Le  Chatelier,  "Re- 
cherches  sur  les  equilibres  chimiques  "  *  (1888),  a 
passage  where  in  speaking  of  a  formula  analogous 
to  equation  ^22\  the  author  makes  the  following 
statement : 

"  It  is  very  probable  that  the  constant  of  integra- 
tion like  the  coefficients  of  the  differential  equation 
is  a  definite  function  of  certain  physical  properties 
of  the  reacting  substances.  The  determination  of 
the  nature  of  this  function  would  lead  to  a  com- 
plete knowledge  of  the  laws  of  equilibrium;  it 
would  make  it  possible  to  determine,  a  priori,  all 
the  conditions  of  equilibrium  relating  to  a  given 
chemical  reaction  without  the  addition  of  new  ex- 
perimental data.  The  exact  nature  of  this  constant 
has  not  been  determined  up  to  the  present  time." 

In  these  words  the  renowned  French  chemist 
not  only  formulated  the  problem  under  discussion 
in  a  very  exact  manner,  but  he  seems  also  to  have 
had  some  idea  of  the  method  for  its  solution  (see 
page  204).f  For  my  part  I  should  like  to  add 
that  the  new  theorem  used  by  me,  and  which,  as  I 
believe  I  have  shown,  leads  to  the  solution  of  the 

*  Also  printed  in  Annales  des  mines  (memoires),  ser.  8,  vol.  13.  See 
page  336. 

f  Corresponding  to  Annales  des  mines  (memoires),  ser.  8,  vol.  13, 
page  356. 


56  THERMODYNAMICS  AND  CHEMISTRY 

problem,  will  perhaps  prove  fruitful  in  other 
branches  of  science  (theory  of  mixtures,  theory  of 
radiation,  etc.). 

In  1902  a  very  interesting  paper  was  published 
by  T.  W.  Richards  *  on  "  The  relation  of  changing 
heat  capacity  to  change  of  free  energy,  heat  of 
reaction,  change  of  volume,  and  chemical  affinity,'' 
in  which  he  pointed  out  very  clearly  that  the  ques- 
tion whether  A  >  Q  or  Q  >  A  above  absolute  zero 
(where  A  =  $),  depends  upon  whether  the  heat 
capacity  is  increased  or  decreased  by  the  chemical 
process,  and  I  am  very  glad  to  be  able  to  state 
that  our  formulas  agree  qualitatively  in  many 
cases  with  the  conclusions  of  Richards.  I  do  not 
wish  to  enter  here  into  a  discussion  of  the  differ- 
ences in  the  quantitative  relations. 

The  point  of  view  taken  by  van't  Hoff  in  the 
"Boltzmann  Festschrift"  (1904)  in  following  up 
the  conclusions  of  Richards,  tends  qualitatively  in 
a  somewhat  similar  direction,  but  quantitatively 
is  very  different. 

Furthermore,  I  wish  to  mention  the  fact  that 
Haber,  in  his  remarkable  book  "  Thermodynamik 
technischer  Gasreaktionen "  (which  appeared  re- 

*  T.  W.  Richards,  Proceedings  of  the  American  Academy  of  Arts 
and  Sciences,  38,  293. 


DETERMINATION  OF  THE  INTEGRATION  CONSTANT   57 

cently),  also  clearly  formulated  the  problem  under 
discussion  and,  at  any  rate  in  some  cases,  at- 
tempted its  solution.  His  deductions,  however, 
appear  to  me  to  differ  from  mine  in  important 
points  ;  in  particular,  in  that  the  integration  con- 
stant according  to  my  formulas  does  not  become 
zero,  as  Haber  considers  it  may,  for  gaseous  re- 
actions in  which  the  number  of  molecules  remains 
unaltered.  Haber,  however,  fully  recognized  the 
importance  of  specific  heats  for  the  further  de- 
velopment of  thermodynamics. 

The  most  important  problem  appears  to  be  the 
numerical  evaluation  of  the  integration  constant  i. 
After  having  solved  this  problem,  we  shall  be 
able  to  test  our  hypothesis  directly  by  calculating 
the  equilibria  of  gaseous  systems.  In  the  second 
lecture  we  deduced  the  equation  (13), 


By  introducing  the  vapor  pressure/*  According  to 
the  formula 

we  obtain 
(30) 


58  THERMODYNAMICS  AND  CHEMISTRY 

My  first  plan  was  to  use  for  this  purpose  the 
theorem  of  corresponding  states,  according  to 
which  the  equation 


in  which  TT  is  the  critical  pressure  and  T  the 
critical  temperature,  must  hold  for  all  substances. 

(T\ 

In  this  equation  f  ( — )  is  a  function  of  the  tem- 
perature, independent  of  the  nature  of  the  sub- 
stance in  question.  As  an  approximation  formula, 
van  der  Waals  derived  the  following : 


(30  „  p 

or 

(31a)        \ogS>=-a^-  +  a  + 


in  which  a  should  have  a  value  constant  for  all 
substances.  Van  der  Waals  found  this  value  to 
be  about  3.0. 

Unfortunately  the  theorem  of  corresponding 
states  is  very  far  from  being  true,  as  has  been 
pointed  out  by  several  authors,  and  this  I  wish  to 
illustrate  by  the  accompanying  curves  (Fig.  8),  in 
plotting  which  the  latest  determinations  for  hydro- 


DETERMINATION  OF  THE  INTEGRATION  CONSTANT    59 

gen,  argon,  etc.,  have  been  used.     It  is  advanta- 
geous to  plot  as  abscissas  the  values  of  I—1  —  1 J, 

and  as  ordinates  the  values  of  log  — ,  as  has  been 

p 


4.U 


M 


3.U 


X'.O 


1.25 


FIG.  8. 


1.  Hydrogen. 

2.  Argon. 

3.  Krypton. 

4.  Oxygen. 

5.  Carbon  bisulphide. 


6.  Phenyl  fluoride. 

7.  Ether.     ^ 

8.  Propyl  acetate. 

9.  Ethyl  alcohol. 


done  in  the  above  figure.     Although  we  obtain 
from  the  formula  (3  la) 


60  THERMODYNAMICS  AND  CHEMISTRY 

curves  which  are  almost  straight  lines,  these  curves 
not  only  do  not  coincide,  but  obviously  diverge 
more  and  more  as  they  approach  absolute  zero. 
We  are  forced  therefore  to  give  up  this  method, 
but  not  without  having  proved  that  the  vapor 
pressure  curves,  even  if  not  coincident,  are  still 
similar  and  would  not  intersect  each  other.* 

Adopting  now  the  thermodynamic  method  we 
may  recall  equation  (30) 


to  which,  as  has  been  stated,  the  gas  laws  are 
supposed  to  be  applicable.  Unfortunately  the 
range  of  temperature  within  which  the  gas  laws 
hold  for  the  saturated  vapor,  and  where  we  have 
experimental  data  applicable  to  the  above  formula, 
is  small,  and  it  is  therefore  impossible  to  deter- 
mine the  coefficients  XQ,  a,  5,  .  .  . ,  i,  even  with  a 
moderate  degree  of  accuracy.  It  consequently 
seems  desirable  to  fill  this  experimental  defi- 
ciency by  continuing  the  vapor  pressure  curves 
as  far  as  possible  in  the  direction  of  very  small 
pressures,  for  example  to  0.001  mm.  of  mercury, 

*Bingham,  J.  Am.  Chem.  Soc.,  28,  717. 


DETERMINATION  OF  THE  INTEGRATION  CONSTANT   61 

and  I  have  begun  to  work  in  this  direction  in  my 
laboratory. 

From  examination  of  equation  (30)  it  would 
not  appear  very  probable  that  the  curves  drawn 
in  the  figure  would  be  so  nearly  straight  lines  as 
the  experimental  data  show,  and  as  equation  (3 la) 
requires.  Since  without  doubt  the  first  members 
are  the  most  important,  partial  mutual  compensa- 
tion is  to  be  expected,  that  is,  the  coefficients  a 
and  b  must  have  opposite  signs.  Since  at  ordi- 
nary temperatures,  the  specific  heat  of  the  liquid 
is  always  greater  than  that  of  the  saturated  vapor, 
at  very  low  temperatures  the  specific  heat  of  the 
saturated  vapor  must  conversely  be  greater  than 
that  of  the  liquid. 

This  conclusion,  drawn  from  the  form  of  the 
vapor  pressure  curves,  is  justified  by  applying  the 
kinetic  theory  of  gases.  According  to  this  theory, 
for  a  monatomic  gas  the  molecular  heat  at  constant 
pressure,  Hp,  must  be  equal  to  5.0  at  all  tempera- 
tures, and  the  molecular  heats  of  polyatomic  gases 
must  always  be  greater  than  5.0 ;  whereas  the 
molecular  heats  of  liquid  or  solid  bodies  as  given 
by  the  latest  measurements  at  low  temperatures, 
diminish  very  rapidly  with  the  temperature.  By 
examining  the  experimental  data  and  by  calcu- 


62  THERMODYNAMICS  AND  CHEMISTRY 

lating  a  large  number  of  vapor  pressure  curves,  I 
arrived  finally  at  the  following  conclusions : 

(1)  The  molecular  heats  of  gases  at  constant 
pressure  near  absolute  zero — at  absolute  zero  itself, 
of  course,  a  gas  cannot  exist  and  must  be  a  con- 
densed crystallized  or  amorphous  body — can  be 
expressed  by 

(32)  JH*  =  3.5  +  m  X  1.5, 

in  which  m  denotes  the  number  of  atoms  in  each 
molecule  of  the  gas. 

According  to  this  equation  the  molecular  heats 
of  the  monatomic  gases, 

H°  —  3.5  +  1.5  =  5.0, 
are  in  agreement  with  the  kinetic  theory. 

For  diatomic  gases  we  have 

H*  =  3.5  +  2  X  1.5  =  6.5, 

whereas  by  extrapolation  of  Langen's*  measure- 
ments for  O2,  Hz,  If2,  CO,  and  for  which  he  gave 
the  expression 

H*  =  6.8  +  0.0012*, 

we  do  in  fact  obtain  for  T—  0  or  t  =  —  273 
^•=6.6. 

The  extrapolation  of  the  new  measurements  of 
Holborn  and  Austin  f  for  OO2  and  of  Holborn 

*  Langen,  Ztschr.  d.  Vereins  deutscher  Ingenieure,  47,  637. 
t  Holborn  and  Austin,  Sitzungsber.  d.  K.  Akad.  d.  Wissensch., 
Berlin,  p.  175  (1905). 


DETERMINATION  OF  THE  INTEGRATION  CONSTANT  63 

and  Hemming*  for  H2O  gives 

fip°  =  7.3  for  CO, ;  Np°  =  7.6  for  #8<9, 
which  seem  to  agree  sufficiently  well,  considering 
the  large  extrapolations,  with  the  value  8.0  calcu- 
lated from  equation  (32). 

Other  measurements  as  well,  perhaps  hardly 
accurate  enough  for  extrapolation,  appear  not  to 
contradict  this  formula. 

(2)  For  the  specific  heats  of  liquids  or  solids  at 
the  absolute  zero,  our  hypothesis  requires  that 
every  atom  shall  have  a  definite  value  for  the 
atomic  heat,  independent  of  the  form,  crystallized 
or  liquid  (i.  e.,  amorphous),  and  of  whether  it  is  in 
chemical  combination  with  other  atoms. 

Numerous  measurements  by  different  experi- 
menters have  shown,  in  full  agreement  with  each 
other,  that  the  atomic  heats  in  the  solid  state 
decrease  greatly  at  low  temperatures,  but  at  the 
present  time  it  is  impossible  to  calculate  the 
limiting  value  toward  which  they  tend.  For 
want  of  a  better  assumption  I  believe  we  can  set 
for  the  present  the  value  of  the  atomic  heats  at 
absolute  zero  for  all  elements  equal  to  1.5.  Of 
course  it  is  somewhat  unsatisfactory  to  calculate 
with  such  a  doubtful  value ;  but  on  the  one  hand 

*  Holborn  and  Hemming,  Drude's  Ann.,  18,  739. 


64  THERMODYNAMICS  AND  CHEMISTRY 

we  are  obliged  for  the  sake  of  the  following  calcu- 
lations to  make  some  assumption,  and  on  the  other 
hand  it  makes  little  difference  for  the  following 
purposes  what  value  the  atomic  heat  has  between 
the  limits  0  and  2.  That  the  atomic  heats,  how- 
ever, do  sink  to  such  small  values  for  the  elements 
like  IfjCj-NyS,  O,  Cl,  is  unquestionable.  It  is 
with  the  compounds  of  these  elements  that  we 
shall  be  concerned  in  our  subsequent  calculations. 

By  combining  the  two  statements  we  find  that 
near  absolute  zero  the  molecular  heat  of  the  vapor 
at  constant  pressure  would  exceed  the  molecular 
heat  of  the  condensed  product  by  an  amount 
equal  to  3.5  gram  calories.  This  statement  seems 
to  me  not  improbable  from  the  point  of  view  of 
the  kinetic  theory. 

(3)  The  theorem  of  corresponding  states,  which 
we  find  to  be  very  far  removed  from  the  truth 
when  applied  to  vapor  pressures,  agrees  very  satis- 
factorily, as  Young  *  has  shown,  for  the  volume 
relations.  Young  states  in  particular  that  the 
volumes  of  the  saturated  vapors  of  the  substances 
investigated  by  him  have  almost  identical  values 
at  corresponding  pressures.  I  found  as  a  very 
simple  empirical  function  the  equation 

*  Young,  Phil.  Mag.  [5],  34,  505. 


DETERMINATION  OF  THE  INTEGRATION  CONSTANT   65 


v  -  v)  =  XT(I  -  L}, 


(33) 

in  which  v  denotes  the  molecular  volume  of  the 
vapor,  v'j  that  of  the  liquid,  p  the  pressure  of  the 
vapor,  and  TT  the  critical  pressure. 

The  following  table  (III)  shows  the  range  of  its 
accuracy  for  phenyl  fluoride  (C6If5f)  which 
Young  *  used  as  his  standard  substance. 

TABLE  III 


T 

P 

V 

v' 

p(v-v>) 

*(>-3 

T 

small 

small 

large 

small 

0.0821 

0.0821 

367.3 

1.316 

22.00 

0.103 

0.0786 

0.0796 

435.0 

6.580 

4.634 

0.115 

0.0684 

0.0700 

473.6 

13.16 

2.265 

0.125 

0.0593 

0.0579 

519.7 

26.32 

1.009 

0.145 

0.0438 

0.0337 

550.0 

39.5 

0.516 

0.179 

0.0242 

0.0094 

559.6 

44.6 

0.270 

0.270 

0.000 

0.000 

The  formula  can  therefore  be  employed  up  to 
rather  high  pressures  if  we  accept  the  above 
statement  of  Young  as  holding  true  in  all  cases. 

The  heat  of  vaporization  can  also  be  expressed 
by  the  formula 

(34)         X  =  (Ao  +  3.5T7-  e?72)  (l  --  ^). 

An  experimental  verification  of  this  formula  will 

*  Loc.  cit. 


66  THERMODYNAMICS  AND  CHEMISTRY 

be  given  later;  if  we  introduce  these  two  last 
equations  in  the  equation  of  Clausius-Clapeyron 


we  have 


dT 
whose  integral  is 

(35)  top^- 


adopting  the  same  notation  for  the  integration 
constant  as  in  equation  (30). 

Since  equations  (33)  and  (34),  though  not  valid 
up  to  the  critical  point,  do  nevertheless  hold  up 
to  a  point  fairly  close  to  it,  it  is  clear  that  the  in- 
tegration of  equation  (35)  with  the  critical  point 
as  one  of  the  limits  of  integration  will  give  an 
equation  which  will  not  be  without  its  uses. 

For  the  desired  integration  constant  i,  or,  more 
simply,  for  the  value  of  <7as  used  in  the  following 
discussion,  equation  (35)  gives  us 


The  following  data  are  therefore  necessary 


DETERMINATION  OF  THE  INTEGRATION  CONSTANT    67 

(1)  A  point  in  the  vapor  pressure  curve  pl9  7J. 

(2)  The  value  of  —  for  T=  T,  (for  the  calcula- 

dT 

tion  of  €).* 

(3)  The  value  of  the  heat  of  vaporization  Xj  cor- 
responding to  the  temperature  T±  (for  the  calcula- 
tion of  X  according  to  formula  (34)). 

(4)  An  approximate  value  for  the  critical  pres- 
sure. 

*  By  differentiating  equation  (34),  noting  that  J!_  is  small  in  com- 
parison  to  1,  we  flnd  .  =  i  (3.5  -  *  +  *£.*). 


LECTUKE  VII 

DETERMINATION  AND  EVALUATION  OF  THE 
CONSTANT  OF  INTEGKATION  BY  MEANS 
OF  THE  CURVE  OF  VAPOR  PRESSURE.- 
(Concluded) 

As  an  example  of  the  method  of  calculating  O 
let  us  consider  the  case  of  ammonia,  which  has 
been  studied  very  thoroughly  by  Dieterici.*  Plac- 
ing p  —  4.21  atmospheres,  T=  273,  X  =  5265, 
TT  —  113  atmospheres,  we  find 


and  consequently,  from  equation  (36), 
O=  3.28 

A  further  control  of  the  value  of  c,  and,  in  gen- 
eral, of  the  reliability  of  this  method  of  calculation, 
is  obtained  from  equation  (34), 


Substituting  in  this  the  above  values 

AO  =  6580,    e  =  0.02785,    TT  =  113, 

*  Dieterici,  Ztschr.  fiir  Kalteindustrie,  1904. 


DETERMINATION  OF  THE  INTEGRATION  CONSTANT   69 

we   obtain   the  values    given   in    the    following 
table : 

TABLE  IV 


p 

T 

A  (calc.) 

x  (obs.) 

4.2 

273. 

(5260) 

(5260) 

8.5 

293. 

4820 

4850 

15.5 

313. 

4270 

4390 

25.8 

333. 

3590 

3870 

48.3 

363. 

2390 

2940 

As  the  table  shows,  equation  (34)  has  a  validity 
similar  to  that  of  equation  (33),  that  is,  it  holds  up 
to  pressures  of  about  20  atmospheres.  The  values 
given  under  X  (obs.)  are  the  heats  of  vaporization 
which  were  found  by  Dieterici  in  the  investigation 
already  quoted. 

Last  summer,  Dr.  Brill*  measured  with  the 
greatest  care  the  vapor  pressures  of  liquid  am- 
monia down  to  the  melting  point.  In  the  follow- 
ing table,  the  observed  values  are  given  together 
with  those  calculated  according  to  equation  (36), 
which  becomes 


6580 


0.02785?7 


*  Brill,  Ann.  der  Physik,  [4]  21, 170. 


70 


THERMODYNAMICS  AND  CHEMISTRY 


TABLE  V 


T 

p  (mm.  Hg)  (obs.) 

p  (calc.) 

195.4 

44.1 

43.9 

200.3 

62.5 

64.7 

204.7 

87.5 

90.8 

210.2 

136.0 

133.9 

216.5 

210.0 

204.6 

222.3 

309.3 

294.1 

228.0 

437.1 

411.5 

234.8 

610.4 

599.2 

240.0 

761.0 

787.0 

In  calculating^?  the  empirical  value 
O=  3.31, 

was  used,  which  is  in  very  satisfactory  agreement 
with  the  value  found  above, 

(7=  3.28. 

By  calculating  a  number  of  examples  I  found 
that  we  can  set  with  sufficient  accuracy 

(37)  -    dX       * 


JiQ  being  the  molecular  heat  of  the  liquid  at  T. 
This  equation,  moreover,  must  be  the  more  exact 
the  lower  the  temperature.  Combining,  further, 
equations  (32),  (36),  and  (37),  we  obtain 


DETERMINATION  OF  THE  INTEGRATION  CONSTANT    71 
1        /  \! 


The  experimental  data  necessary  for  the  appli- 
cation of  (38)  are  more  easily  obtained  than  those 
required  in  order  to  apply  (36). 

In  Table  VI  are  given  the  values  of  C  for  vari- 
ous liquids,  calculated  according  to  formula  (38). 
The  following  table  (Via)  contains  the  values  for 
oxygen  and  nitrogen,  which  are  calculated  in  ex- 
actly the  same  way  as  was  done  in  the  case  of 
ammonia,  using  the  experimental  data  recently 
published  by  Alt.* 

TABLE  VI 


A, 

'SUBSTANCE 

ITi 

Pi 

l± 

A, 

C 

Sulphur  dioxide  .... 
Carbon  bisulphide  .  . 
Chloroform  

273. 
273. 
313. 

1.53 
0.168 
0.487 

6000 
6766 
7490 

20.3 
17.9 
28.3 

3.42 
3.26 

4.07 

Ethyl  ether  

273. 

0.244 

6919 

39.1 

3.56 

Benzol  

293. 

0.099 

8142 

32.2 

3.12 

Alcohol  

303. 

0.103 

10100 

28.5 

4.48 

Water  

273. 

0.0060 

10670 

18.0 

4.26 

*  Alt,  Abh.  d.  Bayr.  Acad.  d.  Wiss.,  II  Kl.,  22  (1905). 


72  THERMODYNAMICS  AND  CHEMISTRY 

TABLE  Via 


A! 

d\ 

SUBSTANCE 

r» 

Pi 

i-* 

~d~T 

Ao 

a 

IT 

0*. 

78 

0.24 

1740 

6.7 

1826 

2.20 

N»  

68 

0.26 

1470 

7.7 

1572 

2.37 

It  may  be  hoped  that  these  values  of  C  are  at 
least  approximately  correct.  A  support  for  their 
correctness  is  the  fact  that  the  calculation  of  <7from 
equation  (38),  which  for  sufficiently  small  vapor 
pressures  is  directly  applicable  to  the  solid  state, 
gives  similar  values.  This  is  shown  in  Table 
VII,  for  which  the  heat  of  vaporization  of  iodine 
was  calculated  from  its  vapor  pressure,  and  the 
value  of  Xj  for  the  other  two  substances  by  the 
addition  of  the  molecular  heat  of  fusion  to  the 
heat  of  vaporization  as  calculated  above. 

TABLE  VII 


SUBSTANCE 

ft 

Pi 

A, 

h» 

C 

Benzol  

273 

0.0322 

10550 

24.6 

3.22 

Water  

273 

0.0060 

12110 

9.0 

3.44 

Iodine  

374 

0.065 

13940 

13.7 

4.04 

During  the  past  summer  Naumann,  a  student 
in  my  laboratory,  made  a  number  of  determina- 


DETERMINATION  OF  THE  INTEGRATION  CONSTANT    73 

tions  of  the  vapor  pressure  of  iodine  at  moderate 
temperatures,  by  measuring  the  quantities  of 
iodine  carried  over  by  a  current  of  hydrogen. 
His  work  is  not  yet  finished,  but  the  following 
values  may  be  given: 

TABLE  VIII 


ATMOSPHERES 

rp        073 

p  (obs.) 

p  (calc.) 

-21. 

0.000004 

0.000005 

19. 

0.00024 

0.00024 

60. 

0.0067 

0.0050 

85. 

0.0267 

0.0222 

137. 

0.263 

0.263 

According  to  the  formulas  and  data  given  above, 
the  vapor  pressure  must  obey  the  equation 

14609  0.0143 


The  agreement  between  the  observed    and   cal- 
culated values  is  sufficient,  if  we  set 
G=  3.925, 

as  has  been  done  in  calculating  p  in  the  above 
table  ;  whereas  we  found  above,  in  a  very  different 
manner, 

(7=  4.04. 


74  THERMODYNAMICS  AND  CHEMISTRY 

Unfortunately  the  experimental  data  which 
would  be  required  in  order  to  apply  equation  (38) 
to  the  solid  or  liquid  state  are  available  for  only  a 
few  substances.  But  the  very  probable  assump- 
tion, that  the  curves  shown  in  Fig.  8  extend  with- 
out intersection  to  as  high  values  of  the  abscissa. 

( — —  ll  as  we  may  choose  to  take,  furnishes  us 

with  a  means  of  determining  by  a  kind  of  inter- 
polation the  value  of  G  for  any  substance  if  we 
know  a  single  point  in  the  curve  of  that  substance. 
This  procedure  is  rendered  very  much  simpler  by 
the  very  evident  fact  that  the  values  of  G  be- 
come larger  the  more  the  curve  for  a  given  sub- 
stance is  inclined  with  respect  to  the  axis  of 
abscissas.  Table  IX  shows  this  clearly ;  together 
with  the  values  of  G  are  given  the  values  for  a  as 
found  from  equation  (31)  for  the  different  sub- 
stances, and  corresponding  to  values  of  —  chosen 

between  1.25  and  1.40.  It  seemed  most  correct 
to  limit  the  comparison  to  the  initial  parts  of  the 
curves  where  formula  (31)  evidently  holds  most 
accurately. 


DETERMINATION  OF  THE  INTEGRATION  CONSTANT    75 
TABLE  IX 


c 

a 

C 

a 

If. 

2.37 

2.4 

Ether  

3.56 

3.0 

•«•»  g  •  •  • 

0,.... 
NH^  .. 

2.20 
3.28 

2.6 
3.0 

Chloroform.  .  .  . 
Benzol  

4.07 
3.15 

2.9 

2.85 

SO. 

3.42 

3.0 

Water  

3.6 

3.3 

CS9. 

3.26 

2.75 

Alcohol  

4.48 

3.7 

"^2     ' 

For  substances  for  which  it  has  apparently 
been  possible  to  determine  C  with  some  degree  of 
accuracy  (the  most  uncertain  are  evidently  oxygen 
and  nitrogen),  we  find  approximately 

(39)  C=l.la. 

Table  X  contains  the  values  of  C  calculated  ac- 
cording to  equation  (39)  for  a  considerable  num- 
ber of  substances.  These  are  the  values  which  we 
shall  use  hereafter  in  the  calculation  of  chemical 
equilibria. 


TABLE  X 


H2  

2.2 

NZ0.... 

3,3 

CHA 

2.5 

•»  jv 
H9S.. 

30 

N*  

....  2.6 

2^  * 

SO*  .... 

3,3 

Oo 

2.8 

CO*  .... 

3fl 

CO. 

3.6 

v^g 

C89 

31 

CL 

3.0 

NHS 

3,3 

/g  

4.0 

H»0.. 

37 

HCl.. 

.  3.0 

•»gw 

COL 

3.1 

Benzol 3.1 

Alcohol 4.1 

Ether 3.3 

Acetone 3.7 

Propyl  acetate .  .  3.8 


NO. . .  (about)  3.7      CHC13 ..  3.2 


76  THERMODYNAMICS  AND  CHEMISTRY 

From  these  figures  we  may  conclude  that  for 
substances  which  do  not  associate,  C  increases 
quite  regularly  with  the  boiling  point,  so  that  we 
can  interpolate  with  a  fair  degree  of  certainty  the 
values  of  O  for  other  substances  which  are  not 
associated ;  whereas  associated  substances  (water, 
alcohol,  acetone,  probably  NO,  perhaps  also  CO) 
have  distinctly  higher  values  than  would  corre- 
spond to  their  boiling  points. 

Systematic  observations,  especially  by  the  ex- 
tension of  the  measurements  of  vapor  pressures 
down  to  very  low  temperatures,  and  also  the  de- 
termination of  specific  heats  between  wide  limits, 
would  therefore  probably  enable  us  to  determine 
with  sufficient  accuracy  the  value,  for  pure  sub- 
stances, of  the  integration  constant,  or,  as  it  may 
appropriately  be  called,  the  chemical  constant  C. 


LECTURE  VIII 

THE  CALCULATION  OF  CHEMICAL  EQUILIBRIA 
IN  HOMOGENEOUS  GASEOUS  SYSTEMS 

FOB  the  calculation  of  the  equilibrium  in  a 
gaseous  system  we  developed  equations  (22)  and 
(29),  which  can  be  combined  to  the  following  : 


Introducing,  as  before,  partial  pressures  in  place 
of  concentrations,  and  letting 


we  have 
(40) 


Moreover,  according  to  (15)  and  (16), 

2*(*  +  £)  =  ^vHpQ  =  2*3.5, 
and  by  equation  (36) 


2.303 


78  THERMODYNAMICS  AND  CHEMISTRY 

Substituting  these  values  we  obtain,  finally, 

fr 


For  the  calculation  of  /?  equations  (15)  and  (32) 
are  employed,  according  to  which 


Hp  =  ffp°  +  2/3T=  3.5 
where  Hp  denotes  the  molecular  heat  at  any  given 
temperature  T,  and  hence 

(42)  S 

since  by  equation  (24) 

S^OQ  =  0. 

Finally,  it  may  be  recalled  that  if  we  represent 
the  heat  of  a  reaction  at  constant  pressure  by  Qp', 
we  have  the  equation 


As  the  first  application  of  the  above  formulas, 
let  us  discuss  the  equilibrium  between  two  optical 
antipodes,  to  which  my  Berlin  colleague  van't 
Hoff  directed  my  attention  in  a  conversation. 
The  equation  of  the  reaction  is 


and  if  pl  and  p/  denote  the  partial  pressures  cor- 


HOMOGENEOUS  GASEOUS  SYSTEMS  79 

ssponding  to  this  homogeneous  gaseous  equilib- 
our  formula  gives 


QQ  is  here  zero,  as  is  well  known,  and,  since 
the  specific  heats  of  the  two  forms  are  exactly 
equal  to  each  other, 

2*(a  +  R)  =  0,    2^/8  =  0,  etc., 
and  as  the  vapor  pressures  are  also  identical, 

^v(i  +  InE)  =  0, 

and  hence  , 

Pi=Pi'- 

Our  theory  yields  therefore  the  well-known  re- 
sult that  in  the  equilibrium  mixture  ("racemic 
mixture  ")  the  concentrations  of  the  two  antipo- 
des are  equal.  This  theorem  has  been  proved 
by  van't  Hoff  by  a  kinetic  method.  Our  theory- 
gives  the  result  in  a  purely  thermodynamic  way. 

To  render  the  trial  of  our  equations  fairly  con- 
clusive they  must  be  applied  only  to  those  equi- 
libria which  have  been  fully  investigated  and  for 
which,  besides  the  thermochemical  data,  the  spe- 
cific heats  and  values  of  G  (chemical  constants) 
are  known.  The  number  of  these  examples  is  not 


80  THERMODYNAMICS   AND  CHEMISTRY 

large,  and  I  have  been  forced  to  use  almost  ex- 
clusively measurements  made  by  myself  and  by 
my  pupils  during  the  last  few  years.  These  re- 
actions are  the  following  :  * 


2. 
3. 
4. 
5. 
6. 

1.  Dissociation  of  Water  Va/por.  —  We  may 
take  here 

for  H20,  H*= 

for  H2  and  O2,     Hp= 

Furthermore, 

Qp'  ==  116000  when  T=  373, 

so  that 

Qp'  =  114600  +  3.5T+0.0012r2, 
and 

log  J^  =  -  ^^  +  1.75  log  T+  0.00028  T-  0.2 
since      &vC=  4.4  +  2.8  -  7.4  =  -  0.2). 

*  To  simplify  the  general  scheme  of  the  reactions,  the  chemical 
equations  are  written  so  that,  as  read  from  left  to  right,  the  reaction 
is  accompanied  by  the  evolution  of  heat. 


HOMOGENEOUS  GASEOUS  SYSTEMS  81 

If  cc  denotes  the  (very  small)  degree  of  dissoci- 
ation at  the  pressure  of  one  atmosphere 

TT" »  Ov 


and 


25050 
T 


+  1.75  log  T+  0.00028  T+  0.1. 


In  the  following  table,  together  with  the  values 
of  the  degree  of  dissociation,  are  given  the  corre- 
sponding temperatures  as  observed  by  Warten- 
berg  and  myself  (loc.  cit.),  and  as  calculated  from 
the  above  equation : 

TABLE  XI 


100  x 

T  (obs.) 

T  (calc.) 

0.0189 
0.199 

1480 
1800 

1455 
1745 

2.  Dissociation  of  Carbonic  Acid. — We  can  set 
forC'0,,  #p=  10.05,) 

for  CO  and  O2,    Hp=6.9,      J> 
Also  since 

Qp'  =  136000  when  T=  290, 

Qp'  =  135300  +  3.5  T-  0.0030  T2, 
and 

log  JT'  =  --  ^^  +  1.75  log  T-  0.00066  T+  3.6 

=  7.2  +  2.8  —  6.4:=  3.6); 


82  THERMODYNAMICS  AND  CHEMISTRY 

and,  exactly  as  in  the  case  of  water  vapor, 


3logcc=  -  +  1.T5  log  T—  0.00066  T+  3.9. 

The  agreement  between  the  calculated  values 
and  those  observed  by  Wartenberg  and  myself  is 
shown  in  the  following  table  : 

TABLE  XII 


100  x 

T  (obs.) 

T  (calc.) 

0.00419 
0.029 

1300 
1478 

1369 
1552 

For  the  ordinary  temperature  (T=  290)  the 
formulas  give 

for  J726>,     100  x  =  10  -*"», 
for  C02,     lOO^^lO-29-35; 

whereas  by  calculating  from  the  measurements  of 
Wartenberg  and  myself  with  the  aid  of  the  second 
law  of  thermodynamics,  we  find 


for  OO       100o?=10-19-1. 


2, 


The  figures  agree  very  well  ;  probably  the  values 
calculated  by  the  purely  thermochemical  method 
for  low  temperatures  are  even  more  exact  than 
the  others. 


HOMOGENEOUS  GASEOUS  SYSTEMS 


83 


3.  Formation  of  Nitric  Oxide.  —  According  to 
Strecker  *  we  have  here 

for 


and  O9,     ?=  1.40;  for  NO,        =  1.39; 

tiv  MV 

and  since    Qp'  =  43200  when  T—  290 
4,'  =  43200  +  0.0004  1  *, 

and     log  JP  =  -  ^r  +  0.00008T+  2.0 

(2if<7=  7.4  -  2.6  -  2.8  =  2.0). 
The  following  table  (XIII)  contains  the  values 
observed  by  me  together  with  those  calculated 
from  the  above  formula.  If  x  denotes  the  volume 
of  NO  formed  per  unit  volume  of  atmospheric  air 
at  the  temperature  in  question,  we  have 


>»-!)M' 

TABLE  XIII 

X 

T  (obs.) 

T  (calc.) 

0.0037 
0.01 

1825 
2205 

1624 
1898 

The  agreement  between  the  observed  and  cal- 
culated values  is  only  approximate ;  but  it  must 

*  Strecker,  Wiedemann's  Annalen,  17, 102. 


84  THERMODYNAMICS  AND  CHEMISTRY 

be  emphasized  that  the  data  for  the  curve  of  va- 
por tension  of  nitric  oxide  are  very  uncertain,  as 
has  also  been  noted  by  Travers,*  who  states  that 
"  the  results  obtained  by  Olszsewski  for  the  vapor 
pressures  of  nitric  oxide  are  somewhat  peculiar." 
In  fact,  the  curve  for  nitric  oxide  plotted  accord- 
ing to  the  method  used  in  Fig.  8  is  very  irregu- 
lar, and  only  permits  the  conclusion  that  it  slopes 
more  steeply  than  the  curves  for  oxygen  and 
nitrogen,  and  therefore  that  nitric  oxide  must 
have  a  distinctly  higher  value  for  C  than  these 
gases.  If,  for  example,  we  were  to  substitute  for 
(7  the  value  3.4  instead  of  3.7  we  should  have  in 
the  above  table  2162  instead  of  1898  as  the  tem- 
perature corresponding  to  x  =  0.01.  A  revision 
of  the  vapor  pressure  of  nitric  oxide  seems  there- 
fore desirable,  and  would  in  itself  be  of  interest 
because  of  the  remarkable  behavior  of  this  sub- 
stance.f 

4.  Formation  of  Hydrochloric  Add. — The  spe- 
cific heats  of  hydrogen  and  hydrochloric  acid  are 
practically  equal  to  each  other;  the  specific  heat 
of  chlorine,  on  the  other  hand,  is  markedly  higher 
(Strecker).  It  is  probable,  however,  that  in  the 

*  Travers,  Experimental  Study  of  Gases,  p.  243. 
f  See  page  76. 


HOMOGENEOUS  GASEOUS  SYSTEMS  85 

case  of  the  specific  heat  of  chlorine  we  are  dealing 
with  irregularities  which  disappear  at  very  low 
pressures.  Since  in  the  following  calculation  (dis- 
sociation of  hydrochloric  acid  at  the  ordinary 
temperature)  only  exceedingly  low  pressures  are 
involved,  and  still  more  because  it  is  in  any  case 
a  question  of  only  a  very  small  correction,  we 
may  set 

2^  =  0, 

and  therefore  2i>8  =  0. 


As  the  heat  of  formation  of  hydrochloric  acid  is 
22000,  we  obtain 

44000 
logJT'  =  logo?2  =  -  -       ^  +  2.2  +  3.0  -  6.0 

4.571  T 
or 

4813 
logoo=  --  —  -  0.4, 

in  which  x  denotes  again  the  (very  small)  degree 
of  dissociation. 

Dolezalek*  has  measured  with  ,  great  care  the 
electromotive  force  of  the  hydrogen-chlorine  cell. 
According  to  the  usual  formulas  this  electromo- 
tive force  can  be  calculated  from 

RT       p 

€  =  -     -  In  -±-  , 
F        TTX' 

*  Dolezalek,  Ztschr.  f.  phys.  Chem.,  26,  334. 


86  THERMODYNAMICS  AND  CHEMISTRY 

in  which  p  denotes  the  partial  pressure  of  the 
hydrogen  and  of  the  chlorine  at  the  electrodes, 
and  TT  the  partial  pressure  of  the  hydrochloric 
acid  over  the  solution  used.  In  the  article  re- 
ferred to  it  is  shown  that  the  electromotive  force 
is  dependent  on  the  partial  pressure  of  the  hydro- 
chloric acid  as  given  in  the  above  formula.  It 
will  suffice,  therefore,  to  calculate  one  value. 
Choosing  for  this  purpose  an  experiment  in 
which  the  concentration  of  the  hydrochloric  acid 
was  six  times  normal,  we  find  in  the  paper  of 
Dolezalek 

TT  =  0.52  mm,     p  =  750  mm, 

and  since  the  temperature  was  30°, 


(log 


=  L168 


while  Dolezalek  gives  the  measured  value  1.160 
volts.  The  agreement  is  excellent,  and  this  ex- 
ample shows  at  the  same  time  how  our  theory 
enables  us  to  calculate  electromotive  forces  from 
thermal  data.  The  calculation  according  to  the 
Helmholtz-Thomson  rule,  by  which  the  chemical 
energy  is  simply  put  equal  to  the  electrical,  gives 
in  this  case,  as  is  well  known,  a  value  which  is 
much  too  great  (about  1.4  volts). 


HOMOGENEOUS  GASEOUS  SYSTEMS 

For  the  equilibria 


87 


and  IT2  +  O12  =  2HCI, 

our  theory  gave 


OF  THE 

UNIVERSITY 


Tr,  9450    . 

log^BT'  =--——+  1.4, 


and 


respectively. 

The  constant  of  integration,  I,  is  positive  in  the 
first  case  and  negative  in  the  second,  which  corre- 
sponds to  the  fact  that  the  vapor  pressure  curve 
of  NO  drawn  as  in  Fig.  8  is  very  much,  that 
of  Hz  very  slightly,  inclined  with  respect  to  the 
axis  of  abscissas.  These  simple  examples  show 
how  different  facts  are  correlated  by  our  theory. 


LECTUEE  IX 

THE  CALCULATION  OF  CHEMICAL  EQUILIBRIA 
IN  HOMOGENEOUS  GASEOUS  SYSTEMS.- 
(  Concluded) 

CONTINUING  the  application  of  our  equations  to 
the  reactions  mentioned  in  the  previous  lecture 
we  may  consider  next 

5.  The  Deacon  Process. 

4HCI  +  O2=2  HZO  +  2  CZ2. 
This  reaction  was  studied  by  Dr.  Vogel  von  Falk- 
enstein,  the  first  account  of  the  work  being  pub- 
lished in  the  Zeitschrift  fivr  Elekt/rochemie* 

We  found  for  the  dissociation  of  water  vapor 


+  0.00028  T+  0.1, 
and  for  the  dissociation  of  hydrochloric  acid, 


log 

~  T 


By  combining  these  two  equations   the  above 

*  Vol.  12,  p.  763. 


HOMOGENEOUS  GASEOUS  SYSTEMS 

equilibrium  can  be  calculated  (as  Bodlander 
pointed  out)  and  we  thus  obtain 


first 


-  0.00028  T-  1.4, 

Table  XIV  shows  the  agreement  between  the 
values  of  K'  found  in  the  very  careful  investiga- 
tion of  Vogel  von   Falkenstein   and   the  values 
calculated  by  the  theoretical  formula. 
TABLE  XIV 


T-273 

K'  (obs.) 

K  (calc.) 

log  K'  (obs.) 

log  K'  (calc.) 

450 
600 
650 

31.0 
0.893 
0.398 

31.9 
0.98 
0.371 

1.49 
-0.050 
-0.400 

1.50 
-0.009 
-0.430 

We  see,  therefore,  that  the  equilibrium  in  the 
Deacon  Process  can  be  calculated  from  our  theory 
very  accurately. 

The  above  examples  show,  first,  that  we  can 
place 

Q,'  =  &' 

without  appreciable  error;  and  secondly,  that  if 
the  temperature  is  not  too  high  the  term  con- 
taining T  as  a  factor  may  be  disregarded.  The 
numerical  calculations  are  thereby  very  much 
simplified. 

*  Bodlander,  Ztschr.  f .  phys.  Chem.,  27,  55. 


90  THERMODYNAMICS  AND  CHEMISTRY 

6.  Formation  of  Hydi*ogen  Sulpliide. 


During  the  past  summer  Dr.  Preuner  has  studied 
this  reaction  in  my  laboratory,  using  the  method 
of  semipermeable  membranes.  He  obtained  the 
following  values  for  the  partial  pressure  of  hydro- 
gen, when  the  total  pressure  of  the  hydrogen  sul- 
phide was  equal  to  one  atmosphere. 

TABLE  XV 


T 

p  (cm.  Hg) 

K' 

1100 

6.5 

4.1  X  10-* 

1220 

11.5 

2.9  X  lO-3 

1320 

16.2 

1.05  X  lO-2 

1420 

20.8 

2.96  X  10~2 

1520 

25.4 

7.5  X  10-2 

Since  sulphur  vapor  is  diatomic  at  these  tem- 
peratures, the  law  of  mass  action  gives 

_. 


Using  the  second  law  of  thermodynamics,  Preuner 
found  for  the  heat  of  reaction  at  constant  volume 
38000  gram  calories.  Since  we  do  not  know  the 
specific  heat  of  hydrogen  sulphide  at  high  tem- 
peratures, we  can  only  use  the  approximate 
formula 


HOMOGENEOUS  GASEOUS  SYSTEMS  91 

1.75  log  ?M-  1.3 
=  4.4  +  2.9  -  6.0  =  1.3). 


The  theoretical  formula  gives  for  K'  =  2.9  X  10  ~  3 
the  temperature  T=  920,  instead  of  1220  as  ob- 
served ;  and  for  J5T'=  2.96  X  10  ~2  the  temperature 
T=  1020,  instead  of  1420  as  observed. 

The  difference  between  the  observed  and  the 
calculated  data  seems  to  indicate  that  the  spe- 
cific heat  of  hydrogen  sulphide  increases  rather 
rapidly  at  high  temperatures,  but  it  must  be 
stated  that  the  above  value  of  Q  may  not  be 
quite  correct. 

Some  of  the  values  given  by  other  experiment- 
ers for  equilibria  in  homogeneous  gaseous  systems 
may  now  be  utilized  for  the  trial  of  our  formulas. 

(7)  Formation  of  Sulphivr  Trioxide  (calculated 
by  Dr.  Falk).  —  The  reaction  is 


Placing 

for  O2,  Hp  =  6.9,  \ 

for  S02,  ffp  =  9.8  (Regnault),  [  at  T=  430, 

for  SOB,  Hp  =  11.0,  J 

=  —  =0.0011. 

860 


92 


THERMODYNAMICS  AND  CHEMISTRY 


The  value  of  Hp  for  SO3  has  never  been  de- 
termined, but  it  seems  reasonable  to  place  it  a  lit- 
tle higher  than  that  of  SO*.  In  any  case,  the 
effect  on  the  final  calculation  is  small,  for  if  we 
should  take  for  80^  Hp  —  11.5,  instead  of  11.0  as 
above,  the  calculated  temperatures  given  in  the 
following  table  (XVI)  would  each  be  increased 
by  only  about  25°. 

Furthermore,  since  Qp'  =  45200  when  T—  290, 
and  therefore  Qp'  =  44100  +  3.5  T+  0.0011T2, 
we  have 

-  ~  +  1.751ogZT+  0.00024T+  3.2 


(2*  (7  =  6.6  +  2.8  -  6.2  =  3.2,  C  for  SO3  being 
put  equal  to  3.1  as  no  experimental  data  for  its 
calculation  are  available.) 


TABLE  XVI 


K 

T  (obs.  by  Bodenstein)* 

T  (calc.) 

0.0053 

852. 

885. 

0.29 

1000. 

1036. 

7.81 

1170. 

1202. 

(8)  Formation  of  Ammonia. — The  formation 
of  this  substance,  the  specific  heat  of  which  is  not 

*  Measurements  of  Bodenstein  quoted  by  Haber,  "  Thermodyna- 
mik  technischer  Gasreaktionen." 


HOMOGENEOUS  GASEOUS  SYSTEMS       93 

definitely  known,  has  been  studied  by  Haber.* 
The  heat  of  formation  of  ammonia  is  about  12000 
gram  calories  per  mol,  and  if  we  consider  an 
equivalent  mixture  of  hydrogen  and  nitrogen,  as 
Haber  does,  denoting  by  x  the  (small)  partial 
pressure  of  the  ammonia  formed,  we  should  have 
0.25  24000 


+  6.6  +  2.6  -  6.6, 
or 

,      0.825  2625       n  _  ,      „  . 

log-—-  =  -  —  -  +  1.75logT+  1.3, 
j. 

from  which  we  find  by  calculation  that  the  value 
x  =  1.2  X  10  ~4  corresponds  to  an  absolute  tem- 
perature of  893,  whereas  Haber  found  experiment- 
ally 1293. 

I  was  astonished  at  this  difference,  and  since 
Haber  only  determined  the  equilibrium  at  one 
temperature,  Dr.  Jellinek,  my  assistant,  and  I  have 
begun  to  investigate  this  equilibrium.  The  val- 
ues which  we  have  found  are,  in  fact,  rather  dif- 
ferent, and  in  much  better  agreement  with  the 
theory.  This  case,  important  in  itself,  will  be 
studied  very  carefully. 

For  an  approximate  calculation   we  may  em- 

•Habev,  "  Thermodynamik  technischer  Gasreaktionen,"  p.  185. 


94  THERMODYNAMICS  AND  CHEMISTRY 

ploy,  as  the  foregoing  results  have  shown,  the 
following  rules  : 

1.  In  place  of  $0'  we  may  substitute  Qp',  the 
heat  of  reaction  at  constant  pressure  at  ordinary 
temperatures,  and  the  term  containing  T  &s  a  fac- 
tor may  be  disregarded. 

2.  If  we  have  no  certain  data  for  the  chemical 
constant  C9  the  average  value  3.0  may  be  used. 

In  this  way  Dr.  Brill*  has  collected  and  calcu- 
lated all  the  available  examples  of  dissociation 
where  the  reaction  follows  an  equation  of  the 
type 

A     _     A   /     I      A  f 
•A-\  —  -O-I    ~T  -"•>  • 

If  cc  denotes  the  degree  of  dissociation  and  P  the 
total  pressure 

JL    ~"~    CC     7~j 


/ 


K'  --  —  —  P 

- 


and  therefore 


For  x  =  0.5  we  obtain  the  approximate  formula 
(42a)   -  log  3  =  -  ^J^  +  1.75  log  T+  3.0  - 

log  P,  in  which  P  is  expressed  in  atmospheres. 

The  following  examples  were  found  in  the  lit- 
erature : 

*  Brill,  Ztschr.  f.  phys.  Chem.,  57,  721. 


HOMOGENEOUS  GASEOUS  SYSTEMS 
TABLE  XVII 


95 


DISSOCIATION  OF 

P(atm.) 

QP' 

T  (obs.) 

T  (calc.) 

N20i  

0.655 

13000. 

323.1 

348.0 

Formic  acid  

1.00 

15000. 

410.0 

410.0 

Acetic  acid  

1.00 

17000. 

430.0 

455.0 

Bromamylenhydrate  . 

« 

Bromine  (a;  =  0.1)  ... 

1.00 
0.10 
0.078 

19500. 
19500. 
59800. 

483.0 
462.0 
1270. 

515.0 
468.0 
1250. 

It  may  be  added  that  measurements  of  the  dis- 
sociation of  iodine  vapor  also  exist,  but  as  they 
were  made  by  the  Victor  Meyer  method  in  which 
the  partial  pressures  are  not  definite,  the  values 
are  not  accurate  enough  for  the  evaluation  of  Q. 

Table  XVIII  gives  the  values  of  Qp'  and  T,  de- 
rived from  equation  (42a),  which  correspond  to  a 
dissociation  of  vc  =  0.5,  and  a  pressure  of  one  at- 
mosphere. 

TABLE  XVIII 


Qf' 

T 

10000 

290 

15000 

405 

20000 

525 

30000 

780 

50000 

1220 

100000 

2350 

200000 

4500 

LECTURE  X 

THE  CALCULATION  OF  CHEMICAL  EQUILIBRIA 
IN  HETEROGENEOUS  SYSTEMS  AND  OF 
ELECTROMOTIVE  FORCES 

IT  can  easily  be  shown  that  the  treatment  of 
heterogeneous  systems  can  be  reduced  to  the  con- 
sideration of  a  homogeneous  system  together  with 
the  various  equilibria  of  vaporization  or  of  sub- 
limation. Let  us  consider  a  reaction  of  the  type 

na  +  v^At  +  v2A2  H  -----  =  vjAJ  -\  -----  , 

in  which  a  denotes  a  species  of  molecule  coexist- 
ing in  the  pure  state  (solid  or  liquid)  with  the 
gaseous  phase.  For  the  gaseous  phase  the  fol- 
lowing equation  holds  : 


-  (n  +  2?)  (i  +  to>E)  —  0. 


On  the  other  hand,  equation  (30)  multiplied  by  n 
gives  for  the  coexistence  of  a  saturated  vapor 
with  its  solid  or  liquid  form 


HETEROGENEOUS  SYSTEMS  97 


-tt 

Subtracting  the  latter  equation  from  the  former, 
all  the  expressions  in  the  first,  third,  and  fifth 
terms  relating  to  the  solid  or  liquid  drop  out, 
a  +  R  and  a  —  a0  +  H  being  both  equal  to  3.5. 
The  second  term  becomes,  as  before,  the  heat  of 
reaction  divided  by  T,  and  for  the  calculation  of 
the  term  which  is  multiplied  by  T  we  have  finally 
the  expression,  similar  to  equation  (42), 
Ti  -f- 


in  which  h  denotes  the  molecular  heat  of  the  spe- 
cies of  molecule  a  in  the  condensed  state  at  the 
temperature  T. 

The  same  equations  evidently  hold  when  any 
number  of  solid  or  liquid  bodies  coexist,  provided 
that  all  are  in  the  pure  state.  We  then  have  the 
general  expression,  analogous  to  equation  (41) 

(43)    log  ^=-—          +  2,1. 


in  which  QQ  denotes  the  heat  developed  by  the 
reaction,  and  also 


98  THERMODYNAMICS  AND  CHEMISTRY 

(44) 


In  calculating  the  remaining  terms  the  species  of 
molecules  A,  present  only  in  the  gaseous  state, 
must  be  taken  into  account.  Examples  to  which 
equation  (43)  can  be  applied  are  very  numerous  ; 
the  calculation  of  ft,  however,  on  account  of  the 
lack  of  exact  data  concerning  the  specific  heats,  is 
possible  in  only  a  few  cases. 

1.  Formation  of  Hydrocyanic  Acid.  —  Dr.  War- 
tenberg  studied  last  summer  the  equilibrium 


If  equal  volumes  of  nitrogen  and  hydrogen  are 
taken,  and  x  denotes  the  fraction  by  volume  of 
the  original  mixture  transformed  into  hydrocyanic 
acid,  we  have 


pl  =  (0.5  —  -J  atmosph 
p2  =  (  0.5  —  -  1  atmosph 


pj  =.  x  atmosphere  ; 
and  therefore 


where  C'  is  the  chemical  constant  for  hydrocyanic 


HETEROGENEOUS  SYSTEMS  99 

acid.     The   following   table  contains  the  results 
obtained  by  Wartenberg : 

TABLE  XIX 


100  z 

T  (obs.) 

T  (calc.)j 

T  (calc.)n 

1.95 

1908 

2174 

1908 

3.1 

2025 

2336 

2032 

4.7 

2148 

2503 

2155 

The  values  T  (calc.)!  were  obtained  by  putting  Gr 
(foTJffCN)  =  4.0,  r(calc.)n  by  putting  &= 4.425. 

A  large  value  for  the  chemical  constant  of  hy- 
drocyanic acid  would  be  expected  from  its  high 
dielectric  constant  (90)  and  *its  great  dissociating 
power,  and  it  is  therefore  probable  that  the  value 
in  question  is  greater  than  that  for  water  (3.7). 

2.  Formation  of  Carbon  Bisulphide. — The  equi- 
librium 


was  measured  by  Koref.  The  heat  of  reaction  at 
high  temperatures  can  be  calculated  from  the  heat 
of  reaction  at  ordinary  temperatures  and  the  dif- 
ference in  energy  between  solid  sulphur  and  sul- 
phur vapor  £>2.  The  latter  is  obtainable  from  the 
experiments  of  Preuner  on  the  heat  of  formation 
of  hydrogen  sulphide  at  high  temperatures,  by 


100 


THERMODYNAMICS  AND  CHEMISTRY 


comparison  with  its  heat  of  formation  at  ordinary 
temperatures.  Placing  2*><? '=  2.9  -  3.1  =  —0.2, 
we  find 

8000 


log 


-0.2. 


4.571  T 

The  following  table  contains  the  percentages  by 
volume  of  OS2  formed  at  different  temperatures, 
together  with  the  temperatures  calculated  from 

our  theory : 

TABLE  XX 


100  x 

T  (obs.) 

T  (calc.) 

1.15 

900. 

1000. 

1.9 

1000. 

1150. 

3.2 

1100. 

1300. 

In  this  case  the  agreement  between  the  observed 
and  calculated  temperatures  would  be  better  if 
Zi'C'were  set  equal  to  zero  instead  of  —  0.2. 

3.  Dissociation  of  Ammo'iiium  Hydrosulpliide. 
— Calculating  the  molecular  heat  of  this  substance 
according  to  Kopp's  method  as  19.1  when  T—  300, 
and  taking  for  the  molecular  heat  of  ammonia  9.5, 
and  for  that  of  hydrogen  sulphide  8.5,  we  obtain, 
since  the  heat  of  dissociation  at  constant  press- 
ure is  22800  at  ordinary  temperatures, 

§'=21900  +  7.0  T-  0.013  T\ 


HETEROGENEOUS  SYSTEMS  101 

Denoting  the  dissociation  pressure  by  P,  we  have 

Z> 


-0.  0014  T+  3.15. 

For  T=  298.1,  the  dissociation  pressure  is  0.661 
atmosphere  ;  from   the   above   equation  the  cor- 
responding temperature  is  calculated  to  be  318. 
With  the  approximate  formula 
P  11400 


the  calculation  gives  a  temperature  of  312.  Since 
the  determination  of  the  coefficient  of  T,  as  re- 
marked several  times,  is  rather  uncertain,  and 
since  the  approximate  formula  obtained  from 
equation  (43)  by  putting 

Qo^Q*,  and  2O/30  +  W3)=0, 
gives  only  slightly  different  values,  we  shall  in 
general  use  this  approximate  formula  in  the  fol- 
lowing examples. 

4.  Vapor  Tension  of  Sodium  Phosphate.  — 
Frowein*  found  the  vapor  tension  of  the  salt 
NaJIPOi  •  12  H20,  for  T=  283.8,  to  be  0.00842 
atmosphere  ;  the  heat  of  hydration,  in  agreement 
with  the  calorimetric  value,  is  calculated  from 
the  change  in  vapor  tension  with  the  tempera- 

*  Frowein,  Ztschr.  f.  phys.  Chem.,  1,  362. 


102  THERMODYNAMICS  AND  CHEMISTRY 

ture  to  be  2244;  and  the  heat  of  dissociation  is 
2244  +  10586  =  12830.  We  have,  therefore, 

1.75  logr  +  3.7, 

from  which  we  find  by  calculation  a  temperature 
of  279  (instead  of  284)  for  the  above  pressure. 
Since  there  are  many  exact  measurements  at  hand 
concerning  the  dissociation  of  salts  containing 
water  of  crystallization,  the  study  of  the  specific 
heats  of  the  salts  in  question  for  the  purpose  of 
testing  the  more  exact  formula  (43)  would  appear 
desirable. 

5.  Revision  of  Trouton's  Rule. — The  empirical 
relation  known  as  "Trouton's  Rule,"  which  is 
formulated 

—  =  constant, 

TQ 

where  X  is  the  molecular  heat  of  vaporization, 
and  T0  the  boiling  point  of  the  substance  on  the 
absolute  scale  of  temperature,  has  been  supposed 
up  to  the  present  time  to  be  at  least  approxi- 
mately correct. 

A  closer  examination  taking  into  account  sub- 
stances having  widely  different  boiling  tempera- 
tures shows,  however,  as  I  wish  to  demonstrate  at 


HETEROGENEOUS  SYSTEMS  103 

this  point,  that  the  above  quotient  is  not  by  any 
means  constant,  but  increases  regularly  with  the 
boiling  temperature. 

In  the  thermodynamic  calculation  of  heats  of 
vaporization,  account  must  be  taken  of  the  fact 
that  at  the  boiling  point,  especially  of  substances 
whose  molecules  are  large,  the  saturated  vapors 
no  longer  strictly  obey  the  gas  laws.  From  the 
formulas  (34)  and  (35)  we  can  obtain 


in  which  p1  and  p2  denote  the  vapor  pressures  cor- 
responding to  T^  and  T2J  two  temperatures  which 
differ  by  so  small  an  amount  that  their  geometrical 
and  arithmetical  means  may  for  practical  pur- 
poses be  set  equal  to  one  another.  This  mean 
temperature  is  the  one  to  which  X  corresponds. 

This  formula  gives  in  fact  values  which  agree 
with  the  direct  measurements  ;  ,  in  general  the 
heats  of  vaporization  calculated  with  its  aid  are 
more  accurate  than  those  determined  calorimetri- 
cally. 

In  Table  XXI  are  given  the  values  for  the  boil- 
ing point  T0,  and  the  heat  of  vaporization  X  at 
this  temperature,  calculated  from  the  above  for- 


104 


THERMODYNAMICS  AND  CHEMISTRY 


mula;  for  hydrogen  alone  the  calorimetrically 
determined  and  very  accurate  value  given  by 
Dewar,  has  been  introduced. 

TABLE  XXI 


SUBSTANCE 

To 

A. 

A. 

To 

9.5  log  T 
-0.007  To 

Hydrogen  

20.4 

248 

12.2 

12.3 

Nitrogen  

77.5 

1362 

17.6 

17.4 

Areron  .  . 

86.0 

1460 

17.0 

17.8 

Oxvffen.  . 

90.6 

1664 

18.3 

18.0 

Methane  

108. 

1951 

18.0 

18.6 

Ethyl  ether  

307. 

6466 

21.1 

21.5 

Carbon  bisulphide  .  .  . 
Benzol  

319. 
353. 

6490 

7497 

20.4 
21.2 

21.6 
21.7 

Propyl  acetate  

375. 

8310 

22.2 

21  8 

Aniline  

457. 

10500 

23.0 

22.1 

Methyl  salicylate  

497. 

11000 

22.2 

22.1 

It  is  evident  that  —  increases  decidedly  and 
^o 

regularly  with  the  boiling  temperature.  The  ex- 
pression given  in  the  last  column, 

^  =  9.5  log  2J-  0.007ro 

J-Q 

(which  I  have  derived  from  certain  considerations 
which  I  shall  not  give  in  detail  here)  agrees  very 
well  with  the  observations,  and  may  perhaps  be 
called  the  "  Revised  Rule  of  Trouton," 


HETEROGENEOUS  SYSTEMS  105 

Substances  which  are  polymerized  in  the  liquid 
state,  but  have  the  normal  vapor  densities  in  the 
form  of  gas,  have  higher  values  for  the  quotient 

-jft  than  are  calculated  from  the  above  formula. 
J-Q 

A  rule  very  similar  to  the  rule  of  Trouton  has 
been  proposed  by  Le  Chatelier  and  further  dis- 
cussed by  Forcrand.*  According  to  this  relation 

0' 

-~  =  constant, 

in  which  Q'  denotes  the  heat  developed  in  the 
dissociation  of  one  mol  of  the  substance  and  1 
the  temperature  at  which  the  dissociation  pressure 
is  equal  to  one  atmosphere.     The  value  of  the 
constant  is  found  to  be  about  33. 

As  an  example  of  this  rule,  we  shall  consider 
6.  The  Dissociation  of  Metal-Ammonia  Chlo- 
rides.— Forcrand  gives  a  resume  of  the  heats  of 
dissociation,  determined  for  constant  pressure  at 
the  ordinary  temperature,  and  the  absolute  tem- 
peratures at  which  the  vapor  tensions  become 
equal  to  the  atmospheric  pressure.  Every  third 
value  has  been  taken  from  the  table  of  Forcrand. 

*  Forcrand,  Ann.  de  chim.  et  de  phys.,  [7]  28,  545. 


106  THERMODYNAMICS  AND  CHEMISTRY 

TABLE  XXII 


Q'  (calories) 

T 

9 

T 

ZnClf\ 

[-6NH 

11000 

332 

33.13 

OaCl2  - 

M-A^s 

10290 

315 

32  66 

2AgCl  - 

|-3JV#3  

11580 

341 

33  96 

PdCL  - 

15560 

483 

3222 

LiCl    - 

11600 

367 

31  61 

i_  g  jyjy8 

11150 

363 

30.72 

3 

0' 

The  expression  ~  is  fairly  constant,  and  Forcrand 

finds  this  confirmed  by  all  the  ammonia  compounds 
investigated.  This  regularity  is  not  only  sugges- 
tive of  Trouton's  Kule,  but  from  the  following 
considerations  the  constancy  appears  here  also  to 
be  limited  to  a  certain  range  of  temperature. 

To  apply  equation  (43)  we  may  set,  according 
to  Kopp,  the  molecular  heat  of  solid  ammonium 
chloride  equal  to  20.0 ;  for  solid  hydrochloric  acid 
the  value  8.6  may  be  calculated,  so  that  for  solid 
ammonia  11.4  remains.  Substituting  for  the  mo- 
lecular heat  of  gaseous  ammonia  9.5,  we  have, 
from  equation  (44) 

Q'=Q0  +  3.5 T-  0.009 T*, 
and  further 


logp  =  - 


4.5711 


-  -  1.75  log  T—  0.002T+  3.3, 


HETEROGENEOUS  SYSTEMS  107 

from  which,  putting  p  =  1,  we   obtain   for    the 

Of 

coefficient  •— 


+  (1.75  log  T  -  0.002  T  +  3.3)  4.671. 

The  second  member  reduces  for  T—  290,  to  the 
value  33.2  ;  for  T  —  358,  to  32.9.  We  see,  then, 

Qr 

that  as  given  by  this  equation,  ~-  must  necessarily 

be  constant  over  a  rather  wide  range  of  tempera- 
ture in  agreement  with  the  empirical  rule  of  Le 
Chatelier-Forcrand  ;  and  that  the  value  which  we 
have  calculated  theoretically  does  in  fact  agree 
completely  with  the  average  value  obtained  from 
Forcrand's  data,  that  is,  32.33  at  an  average  tem- 
perature of  358. 

In  general,  our  theory  furnishes  as  an  approx- 
imate formula  for  the  dissociation  pressure  p  (de- 
noting as  above  by  Q'  the  heat  of  dissociation  per 
mol  at  constant  pressure  and  the  ordinary  tem- 
perature) the  following  : 

(45) 


Since  the  values  of  C  lie  in  the  vicinity  of  3.0,  we 


108  THERMODYNAMICS   AND  CHEMISTRY 

obtain,  independently  of  the  nature  of  the  sub- 
stance in  question,  the  values 

^  =  29.7  when  T=    100, 

"  =  33.6  "  T=  300, 
"  =  35.3  "  T  =  500, 
"  =  37.7  "  T=  1000. 

In  this  we  easily  recognize  the  rule  of  Le  Chatelier- 

Qr 

Forcrand  (according  to  which  the  value  of  —-  is 

about  33  for  widely  different  gases),  but  with 
this  difference,  that  the  hitherto  entirely  em- 
pirical coefficient  33  has  acquired  a  simple 
meaning,  4.57  (1.75  log  T+  <7),  and  that  the 
reliability  of  the  rule  is  obviously  limited  to  a 
middle,  though  rather  extended,  range  of  tem- 
perature. 

In  fact,  if  we  calculate  the  temperature  at  which 
the  dissociation  pressure  is  equal  to  one  atmosphere, 
for  cases  in  which  that  temperature  is  very  high, 
we  find  the  rule  of  Le  Chatelier-Forcrand  does 
not  hold,  but  that  our  theory  gives  a  satisfactory 
agreement,  as  is  shown  by  the  following  table  col- 
lected by  Dr.  Brill : 


HETEROGENEOUS  SYSTEMS 
TABLE  XXIII 


109 


SUBSTANCE 

Q' 

T  (obs.) 

TJL 

33. 

nf  rom  \ 
oq.  (45>; 

OBSERVER 

Ag2C03  .. 

20060 

498 

608 

498 

.  .  Joulin 

PbC03  ... 

22580 

575 

684 

575 

.  .  Colson 

CaC09  .  .  . 

42520 

1098 

1290 

1098 

....  Brill 

SrCO^.... 

55770 

1428 

1688 

1428 

....  Brill 

It  seems  very  remarkable  that  these  values  which 
are  obtained  from  equation  (45)  by  putting^?  =  1, 
and  G=  3.2  for  C02  should  agree  entirely  with 
the  observed  values ;  while  those  in  the  fourth 
column  calculated  according  to  the  rule  of  Le 
Chatelier-Forcrand  show  a  poor  agreement. 

Moreover,  if  we  consider  the  evolution  of  CO 
in  the  well-known  reaction  of  the  formation  of 
calcium  carbide 

CaC2  +CO=  CaO  +  3C+  94750  cal., 
Qf 

we  find  for  the  quotient  —.,  according  to  the  in- 
vestigation of  Rothmund  * =  46.8.  that  is, 

2040 

a  much  higher  value  than  33,  the  one  given  by  the 
above  rule,  but  in  better  accordance  with  the  ap- 
proximate value  4.57(1.75  log  2040  +  3.6)  =  42.9. 

*  Rothmund,  Gottinger  Nachrichten,  1901,  Heft  3. 


110  THERMODYNAMICS  AND  CHEMISTRY 

DETERMINATION  OF  THE  STABILITY  OF  CHEMICAL 
COMPOUNDS 

The  question  whether  a  chemical  compound 
can  be  formed  to  an  appreciable  extent  under 
given  conditions,  is  identical  with  the  question  of 
its  stability.  This  question  can  be  answered  by 
the  formulas  developed  here  with  the  aid  of  the 
heats  of  reaction.  Since  in  general,  under  given 
experimental  conditions,  chemical  compounds  are 
either  very  stable  or  very  unstable,  owing  to  the 
fact  that,  especially  at  low  temperatures,  chemical 
equilibria  in  which  all  the  components  coexist  in 
appreciable  concentrations  are  the  exception  rather 
than  the  rule,  our  formula  even  in  the  approximate 
form  will  generally  give  a  sufficiently  definite 
answer. 

We  have  already  seen  that  ammonia,  for  in- 
stance, which  doubles  in  volume  when  it  dis- 
sociates at  constant  pressure,  is  unstable  at  mod- 
erately high  temperatures  notwithstanding  that 
its  heat  of  formation  is  by  no  means  small.  Ozone 
must  be  unstable  at  low  temperatures,  and  only 
able  to  coexist  to  an  appreciable  extent  with  ordi- 
nary oxygen  at  very  high  temperatures,  because 
its  heat  of  formation  is  negative,  and  it  dissociates 


STABILITY  OF  COMPOUNDS  111 

with  increase  in  volume.  The  halogen  hydrides 
dissociate  without  change  of  volume ;  here  a  con- 
siderable heat  of  formation  corresponds,  at  least  at 
low  temperatures,  to  great  stability  (HCl,  HBr) ; 
hydriodic  acid,  on  the  other  hand,  which  is  formed 
from  its  components  in  the  gaseous  state  without 
any  marked  thermal  effect,  exists  at  low  tempera- 
tures in  a  state  of  equilibrium  with  appreciable 
quantities  of  its  products  of  dissociation.  Evi- 
dently the  task  of  working  over  the  whole  field 
of  chemistry  from  the  point  of  view  of  thermo- 
chemistry lies  before  us.  In  the  field  of  carbon 
compounds  in  particular,  where  the  question  of 
stability  has  remained  almost  wholly  unanswered 
up  to  the  present  time,  much  light  is  to  be  ex- 
pected. Thanks  to  the  determinations  of  the 
heats  of  combustion,  the  thermochemistry  of  or- 
ganic compounds  is  very  accurately  known,  and  a 
thorough  investigation  of  the  question  of  stability 
is  thus  made  possible. 

To  mention  only  a  few  examples,  let  us  recall 
first  that  the  formation  of  gaseous  ethyl  acetate 
and  water  from  alcohol  vapor  and  acetic  acid  va- 
por takes  place  without  the  development  of  an  ap- 
preciable amount  of  heat.  Since  the  reaction  goes 
on  without  change  of  volume,  an  equilibrium  must 


112  THERMODYNAMICS  AND  CHEMISTRY 

be  reached  at  which  all  the  components  are  pres- 
ent in  appreciable  concentrations.  Moreover,  since 
the  vapor  pressures  of  the  four  substances  do  not 
differ  very  much  from  each  other,  this  equilibrium 
must  also  exist  in  the  liquid  mixture.  This  will 
be  recognized  as  the  classic  example  of  a  chemical 
equilibrium,  studied  by  Berthelot. 

For  the  formation  of  acetylene  from  carbon  and 
hydrogen  we  have  the  equation 


<•> 

and  for  benzol  the  corresponding  equation 


Only  at  very  high  temperatures  would  the  right 
side  of  equation  (a)  have  the  value,  for  example, 
of  3.0,  that  is,  only  then  would  0.1  per  cent  by 
volume  of  acetylene  be  stable  in  the  presence  of 
hydrogen  at  atmospheric  pressure.  This  corre- 
sponds evidently  to  the  well-known  formation  of 
acetylene  when  an  electric  arc  is  formed  between 
carbon  electrodes  in  an  atmosphere  of  hydrogen. 
Benzol  vapor,  on  the  other  hand,  in  the  presence 
of  hydrogen  and  solid  carbon,  has  no  evident 
"right  to  exist"  in  appreciable  amounts. 


ELECTROMOTIVE   FORCES  113 

Multiplying  equation  (a)  by  3  and  subtracting 
(b)  we  have 


This  equation  shows,  in  agreement  with  experi- 
ment, that  acetylene,  except  at  extremely  high 
temperatures,  may  polymerize  with  the  formation 
of  benzol. 

It  would  appear  then  that  the  exceptionally 
rich  field  of  the  equilibrium  between  carbon,  hy- 
drogen, and  the  various  hydrocarbons,  would  fur- 
nish ample  opportunity  for  the  application  of  our 
theory. 

CALCULATION  OF  ELECTROMOTIVE  FORCES 

It  is  evident  that  the  calculation  of  the  change 
in  free  energy  by  means  of  thermal  data  also  en- 
ables us  to  calculate  electromotive  forces,  as  was 
shown  in  the  eighth  lecture.  We  shall  now  con- 
sider a  few  additional  examples : 

1.  The  Oxygen-Hydrogen  Cell. — The  second  law 
of  thermodynamics  gives  us  the  relation 

T>rrt  -I 

(46)  £=^W_L_> 

in  which  TTI  and  7r2  denote  the  partial  pressures  of 


114  THERMODYNAMICS  AND  CHEMISTRY 

hydrogen  and  oxygen  in  saturated  water  vapor. 
Using  the  formula  developed  in  the  fourth  lect- 
ure, we  find  for  the  dissociation  of  water  vapor 
referred  to  atmospheric  pressure  at  T=  290° 
(t=  17°  C),  »=  0.48  X  10-25  per  cent.  Reduced 
to  the  tension  of  water  vapor  for  this  temperature, 
_  0.48  X  10~25_  _25 

V  0.0191 
from  which 
TTJ  =  0.0191  X  1.80  X  10"27  atmospheres,     and 

0.0191  X  1.80  X  10~27 

7T2  =  —  atmospheres. 

2 

We  find,  therefore, 

e  =  0.01438  log  4.92  X  1085 
=  1.2322  volts  at  17°  C. 

This  value  is  somewhat  higher  than  the  one 

found  by  direct  measurement  (1.15  volts),  but  it 

seems  certain  that  the  experimental  result  is  too 

low,  owing  to  the  fact  that  the  oxygen  electrode 

never  becomes  completely  saturated  with  oxygen.* 

Without  using  the  values  found  by  the  direct 

experimental  determination  of  the  dissociation  of 

water  vapor,  we  calculated  in  the  eighth  lecture 

from  thermochemical  data  the  value 

x  =  10-25-81. 

*  Nernst  und  Wartenberg,  Ztschr.  f .  phys.  Chem.,  56,  544. 


ELECTROMOTIVE  FORCES  115 

Introducing  this  in  equation  (46),  we  find 

€  =  1.231  volts, 

in   excellent    agreement   with    the   value   found 
above. 

A  more  direct  method  is  based  upon  equation 
(43).  For  the  example  in  question  this  formula 
becomes 

log  K  =  log  Tr2^  -  -  ^J_  +  2*  1.75  logT 

+  Snft  +  2y0 

4.571 

Expressing  $,  the  heat  of  formation  at  constant 
pressure  of  two  molecules  of  liquid  water,  by  the 
relation 

Q  =  137400  +  10.5  17-  0.044T2, 

which  fulfills  the  conditions 

q  =  2  X  68400  when  T=  290, 


=  3  X  3.5  when  T=  0,  and 
dT 


-  =  -  15.6  =  3  X  6.8  -  36  when  T=  290. 
dT 

We  have,  therefore, 

log  ir^ir,  =  --^^  +  5.25  log  T-  0.010  T+  7.2 
(2»<7=  2  X  2.2  +  2.8  =  7.2), 


116  THERMODYNAMICS  AND  CHEMISTRY 

and  consequently 

c  =  0.01438  X  86.39  =  1.242  volts  at  17°  C. 

The  difference  between  this  value  and  the  one 
found  above  (1.231  volts)  is  accounted  for  by  the 
fact  that  the  assumption  upon  which  this  last 
method  of  calculation  is  based,  namely,  that  the 
molecular  heat  of  liquid  water  increases  linearly 
with  the  temperature,  is  evidently  not  accurate. 
There  is,  however,  a  very  simple  expedient  by 
means  of  which  we  can  avoid  this  difficulty.  If 
we  calculate  the  electromotive  force  of  the  oxygen, 
hydrogen  cell  for  T=  273,  the  temperature  at 
which  ice  and  liquid  water  coexist,  the  affinity  of 
the  reaction 

2JZ,  +  O2  =  2N2O  (ice) 

will,  at  this  temperature,  be  identical  with  the 
electromotive  force.     For  this  reaction 

Q  =  138130  +  10.5T-  0.015T2, 
this  equation  fulfilling  the  condition 

Q  =  137000  +  36  X  80  =  139880,  for  T=  273. 
We  find  further 


~         3  V  3  Pi    for   T  —  0 
—  =  t5  x  3.0   ior  ^    -  u, 

<Z-Z 


^.  =  3  X  6.7  -  18  =  2.1  for  T=  273 
dT 


ELECTROMOTIVE   FORCES  117 

(18.0  =  the  heat  capacity  of  two  niols  of  ice), 
and  finally 

logTr^TTj,  = — ^-  +  5.25  log  T—  0.0032  T+ 7.2. 

This  equation  gives  for  T  =  278 

Iog7r18wa=  —  91.56, 
and  therefore 


4 

To  find  the  value  of  this  expression  for  T=  290, 
we  apply  the  well-known  equation 

Q    .^ 


23046          dT 


to  the  liquid  water  cell,  from  which  we  find  for 
ordinary  temperatures 

*~  0.00085. 


Using  this  value  we  obtain 

JS=  1.225  volts  at  T=  290. 
This  value  agrees  better  with  the  first  and  second 
values  calculated  than  with  the  third. 

2.  The  Clcvrk  Element.  —  The  reaction  produc- 
ing the  current  in  the  Clark  element  is  repre- 
sented by  the  equation 
Zn  +  fffrSO*  +  IH^O  =  ZnJSO*  •  IH^O  +  2ffg. 

Our  fundamental  theorem  is  not  directly  ap- 


118  THERMODYNAMICS  AND  CHEMISTRY 

plicable  to  this  reaction,  because,  as  Cohen  *  first 
clearly  showed,  the  zinc  sulphate  formed  passes 
into  solution,  combines  with  the  water,  and  then 
as  a  result  of  the  supersaturation  so  produced 
crystallizes  out,  carrying  down  further  quantities 
of  salt  with  it.  If,  however,  ice  is  one  of  the 
solid  substances  present  in  the  cell,  that  is  to  say, 
if  we  study  the  Clark  element  at  the  cryohydric 
point,  the  reaction  takes  place  between  pure  sub- 
stances, to  which  equations  (23)  and  (24)  are 
directly  applicable. 

Since  the  electromotive  force  at  the  cryohydric 
point  of  zinc  sulphate  ( —  7°  C.)  is  1.4624  volts, 
we  have 
A  =  2  X  23046  X  1.4624  =  67405,   for  T=  266. 

The  evolution  of  heat  at  17°  C.,  the  temperature 
to  which  the  thermal  data  refer,  is  then 
Q  =  66600  (for  T=  290), 

in  which  the  value  88.7  is  assumed  for  the  latent 
heat  of  fusion  of  ice  at  this  temperature,  this 
value  being  derived  from  the  latent  heat  of  fusion 
at  0°  C.  and  the  difference  between  the  specific 
heats  of  ice  and  water.  Under  these  conditions, 
therefore,  A  and  Q  only  differ  by  a  small  amount, 
as  would  follow  also  from  equations  (6)  and  (7), 

*Cf.  W.  Jaeger,  Normalelemente,  Halle,  1902. 


ELECTROMOTIVE   FORCES  119 

in  view  of  the  well-known  fact  that  the  molecular 
heats  of  solid  substances  are  approximately  addi- 
tive. For  the  ordinary  Clark  element,  in  which 
the  behavior  of  the  solution  also  comes  into  con- 
sideration, there  is  certainly  quite  a  large  difference 
between  the  electrical  and  heat  energies.  For  ex- 
ample, for  T=  291,  A  =  65875  and  #=81130. 

Our  new  hypothesis,  therefore,  allows  us  to 
calculate  the  electromotive  forces  of  galvanic  ele- 
ments in  the  following  general  way:  The  gal- 
vanic combination  in  question  is  assumed  to  be 
varied,  using  ice,  if  necessary,  as  one  of  the  solid 
substances  present,  so  that  only  perfectly  pure 
substances  (as  distinct  from  mixtures  or  solutions) 
enter  into  the  equation  representing  the  reaction 
which  produces  the  current.  Knowing  the  heat 
evolved  and  the  specific  heats,  the  coefficients  of 
equation  (23)  can  be  calculated,  and  consequently 
A  and  Q  also.  By  applying  the  well-known  laws 
which  govern  the  change  in  electromotive  force 
with  the  concentration  in  dilute  solutions  (the  so- 
called  osmotic  theory  of  current  production)  the 
electromotive  force  for  any  concentration  can  be 
calculated. 

In  the  equation 

A  =  n  -  E-  23046  =  Q0  -  T*  2^/30, 


120  THERMODYNAMICS  AND  CHEMISTRY 

the  coefficient  2^/30  appears  in  general  to  be 
small.  Its  influence  is  therefore  often  negligible 
at  ordinary  temperatures,  especially  when  the 
electromotive  forces  dealt  with  are  not  too  small. 

CONSIDERATION    OF   THE  KINETIC    BASIS  OF  THE 
NEW  THEOREM 

The  behavior  of  substances  in  the  ideal  gaseous 
state  is,  as  is  well  known,  of  a  very  simple  nature, 
which  has  found  in  the  kinetic  theory  a  theoreti- 
cal explanation.  The  heat  theorem  considered  in 
these  lectures  makes  it  appear  probable  that  also 
in  the  liquid  and  solid  states  at  very  low  tempera- 
tures matter  obeys  strikingly  simple  laws,  and  it 
may  be  hoped  that  in  this  way  new  points  of 
view  have  been  furnished  for  the  development  of 
the  molecular  theory. 

If  we  now  wish,  in  concluding  our  discussion, 
to  take  up  briefly  the  question  of  the  interpreta- 
tion, from  the  standpoint  of  the  molecular  theory, 
of  the  two  equations 


when  T=  0, 


KINETIC  THEORY  121 

it  is  obvious  that  the  first  equation  merely  re- 
quires that  in  the  neighborhood  of  the  absolute 
zero,  the  molecular  heat  of  a  compound  shall  be 
equal  to  the  sum  of  the  atomic  heats  of  the  atoms 
composing  the  compound.  That  is,  every  atom  of 
a  particular  element  requires  the  same  amount  of 
heat  to  produce  the  same  rise  in  temperature,  inde- 
pendently of  the  state  of  aggregation,  crystallized 
or  amorphous,  of  the  substance,  and  also  of  the 
nature  of  the  other  elements  with  which  the  atom 
of  the  element  in  question  may  be  combined. 
The  interpretation  of  the  equation 

lim.—  =  0  when  T=0 
dT 

from  the  kinetic  standpoint  is  more  difficult. 
Since  at  the  absolute  zero  the  kinetic  energy  is 
zero,  the  maximum  work  is  evidently  given  by 
the  sum  of  the  differences  of  the  potential  ener- 
gies which  the  reacting  atoms  possess  before  and 
after  the  reaction.  By  the  motion  of  the  atoms, 
which  corresponds  to  a  definite  elevation  of  the 
temperature  above  absolute  zero,  these  potential 
energies  are  evidently  changed.  The  above  equa- 
tion requires  that  this  change  shall  be  either  in- 
finitely small,  or  independent  of  the  state  in  which 
the  atom  exists. 


122  THERMODYNAMICS  AND  CHEMISTRY 

These  considerations  render  it  very  probable 
that,  as  with  the  specific  heats,  the  expansion  by 
heat  in  the  vicinity  of  absolute  zero  follows  very 
simple  laws.  This  indeed  seems  to  be  the  case, 
as  shown  by  the  empirical  relation  discovered  by 
Tammann,* 

T  — 

J-n  — 


"ro-ro' 

(A-V  denoting  the  change  in  volume  at  the  melt- 
ing point  T0,  y0  and  y/,  the  coefficients  of  expan- 
sion at  T0  of  the  two  coexisting  phases).  This 
equation  is  in  complete  analogy  with  the  relation 

(28)  T0  =  — ^— - 


(which  we  have  found  to  be  a  consequence  of 
equations  (a)  and  (Z>)),  and  it  appears  not  to  be 
improbable  that,  corresponding  to  the  relations 
represented  by  equations  (a)  and  (£),  the  relation 

(c)  —  =  0  when  T=0 

d  JL 

also  holds;  that  is,  the  expansion  by  heat  of 
amorphous  or  crystallized  substances  in  the  neigh- 
borhood of  the  absolute  zero  is  a  purely  additive 
property. 

*  Krystallisieren  und  Schmelzen,  p.  42. 


KINETIC  THEORY  123 

These  observations  may  suffice  to  show  that 
the  further  application  of  the  kinetic  theory  to 
the  behavior  of  solid  and  liquid  substances  at 
temperatures  close  to  the  absolute  zero  promises 
to  yield  fresh  sources  of  information. 


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